# Magnetic field

> Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Magnetic_field
> Markdown URL: https://mediated.wiki/source/Magnetic_field.md
> Source: https://en.wikipedia.org/wiki/Magnetic_field
> Source revision: 1356614648
> License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/)

Property of space that quantifies the magnetic influence at a given location

This article is about the property of space that quantifies the magnetic influence at a given location. For a description of the phenomena of magnetism including different types of magnetic material, see [magnetism](/source/Magnetism).

Electromagnetism Electricity Magnetism Optics History Computational Textbooks Phenomena Electrostatics Electric charge Charge density Coulomb law Electric field Electric flux Gauss's law Electric potential Potential energy Conductor Electret Electric dipole Electrostatic discharge Electrostatic induction Insulator Permittivity Polarization Static electricity Triboelectricity Magnetostatics Magnetic field Ampère's law Ampère's force law Biot–Savart law Gauss's law for magnetism Magnetic dipole Magnetic flux Magnetic scalar potential Magnetic vector potential Magnetization Permeability Right-hand rule Electrodynamics Maxwell's equations Displacement current Electromagnetic field Lorentz force Retarded potentials Liénard–Wiechert potential Jefimenko's equations Radiation Cyclotron radiation Faraday's law Eddy current Induction Lenz's law Electromagnetic mass Abraham–Lorentz force Larmor formula Poynting's theorem Maxwell tensor Electrical network Alternating current Capacitance Current density Direct current Electric current Electric power Electrolysis Electromotive force Impedance Inductance Joule heating Kirchhoff's laws Network analysis Ohm's law Parallel circuit Resistance Resonant cavities Series circuit Voltage Watt Waveguides London equations Magnetic circuit AC motor DC motor Electric machine Electric motor Gyrator–capacitor Induction motor Linear motor Magnetomotive force Permeance Reluctance (complex) Reluctance (real) Rotor Stator Transformer Covariant formulation Electromagnetic tensor Electromagnetism and special relativity Four-current Four-potential Mathematical descriptions Maxwell equations in curved spacetime Relativistic electromagnetism Stress–energy tensor Synchrotron radiation Bremsstrahlung Scientists Ampère Arago Biot Coulomb Davy Einstein Faraday FitzGerald Fizeau Franklin Galvani Gauss Gibbs Gilbert Green Heaviside Helmholtz Henry Hertz Hopkinson Joule Kelvin Kirchhoff Larmor Lenz Liénard Lorentz Maxwell Neumann Ohm Ørsted Poisson Poynting Ritchie Savart Singer Steinmetz Tesla Thomson Volta Weber Wiechert v t e

In [magnetism](/source/Magnetism) and [electromagnetism](/source/Electromagnetism), **magnetic field** is a physical property of space that quantifies the magnetic influence at a given location. Magnetic fields deflect moving [electric charges](/source/Electric_charges) (including [electric currents](/source/Electric_currents)), apply torques on [magnets](/source/Magnet) to twist them in the direction of the magnetic field, and attract or repel magnets and magnetic material such as [iron](/source/Iron). In addition, a time varying magnetic field [induces](/source/Electromagnetic_induction) electrical currents. Magnetic fields are created by magnetic materials and by moving electric charges (including electrical current). The latter is important in creating [electromagnets](/source/Electromagnet): devices that precisely control magnetic fields by changing the current through the electromagnet.

Magnetic fields are used throughout modern science and technology. In [electrical engineering](/source/Electrical_engineering) and [electromechanics](/source/Electromechanics) it is important in the design and use of [electric motors](/source/Electric_motor), [generators](/source/Electric_generator), [transformers](/source/Transformer), electromagnets, and [inductors](/source/Inductor) among many other devices. In [material science](/source/Material_science), magnetic forces give information about the charge carriers in a material through the [Hall effect](/source/Hall_effect) in addition to other uses. In [geology](/source/Geology) and [geophysics](/source/Geophysics), [Earth's magnetic field](/source/Earth's_magnetic_field) gives information about earth's interior while local magnetic field measurements are used in [mineral exploration](/source/Mineral_exploration) and other measurements. Too, Earth's magnetic field creates a [magnetosphere](/source/Magnetosphere) which shields the Earth's [ozone layer](/source/Ozone_layer) and the rest of the planet from the [solar wind](/source/Solar_wind). In [physics](/source/Physics) the relationship between the magnetic and electric fields forms the field of [electrodynamics](/source/Electrodynamics) which is important to understand a wide range of phenomena including [light](/source/Light) (also known as [electromagnetic radiation](/source/Electromagnetic_radiation)) and the properties of [antenna](/source/Antenna_(radio)) and [transmission lines](/source/Transmission_line).

Since both strength and direction of a magnetic field may vary with location, it is described mathematically by assigning a [vector](/source/Euclidean_vector) to each point of space, making it a [vector field](/source/Vector_field).[note 1] There are two different, but closely related, vector fields which are called "magnetic field". These are written as **B** and **H**.[note 2] While the best names for these fields is the subject of long running debate, the underlying physics is uncontested.[1]

## Definitions

See also: [List of electromagnetism quantities](/source/List_of_electromagnetism_quantities)

The international [ISO 80000-6](/source/ISO_80000-6) standard defines **magnetic field** as "that component of an electromagnetic field which is characterized by the magnetic field strength vector **H** and the magnetic flux density vector **B**."[2] This standard also defines **B** and **H** as given in the sections below. While there is wide agreement on these definitions of **B** and **H**, there are many alternative names for both (see sidebars in the corresponding sections).

### The B-field

Magnetic flux density (B)[3] Other names Magnetic induction[4] Magnetic field[5] Common symbols B SI unit tesla (T) Other units gauss γ (nT) lines per square inch In SI base units kg A-1 s-2 Dimension T-2MI-1

See also: [tesla (unit)](/source/Tesla_(unit))

Also known as **magnetic flux density**, the magnetic **B** field causes magnetic forces, magnetic torques and electromagnetic induction. Therefore, it can be defined by any equation that describes these phenomena.

For example, the magnetic field vector **B** at any point can be defined as the vector field that, when used in the [Lorentz force law](/source/Lorentz_force_law), correctly predicts the force on a moving charged particle at that point:[6][7]

**Lorentz force law**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = q E + q ( v × B ) {\displaystyle \mathbf {F} =q\mathbf {E} +q(\mathbf {v} \times \mathbf {B} )}

Here **F** is the force on the particle, *q* is the particle's [electric charge](/source/Electric_charge), **E** is the external electric field, **v**, is the particle's [velocity](/source/Velocity), and × denotes the [cross product](/source/Cross_product).

In other words,[8]

[T]he command, "Measure the direction and magnitude of the vector **B** at such and such a place," calls for the following operations: Take a particle of known charge *q*. Measure the force on *q* at rest, to determine **E**. Then measure the force on the particle when its velocity is **v**; repeat with **v** in some other direction. Now find a **B** that makes the Lorentz force law fit all these results—that is the magnetic field at the place in question.

For more details see [Lorentz Force](/source/Lorentz_Force) or the [#Magnetic force on a charged particle](#Magnetic_force_on_a_charged_particle) section below.

The [SI](/source/SI) unit of **B** is [tesla](/source/Tesla_(unit)) (symbol: T).[note 3] The [Gaussian-cgs unit](/source/Gaussian_units) of **B** is the [gauss](/source/Gauss_(unit)) (symbol: G).[9] (The conversion is 1 T ≘ 10000 G.[10][11]) One nanotesla corresponds to 1 gamma (symbol: γ).[11]

### The H-field

Magnetic field strength (H)[12] Other names Magnetic field intensity[4] Magnetic field Magnetizing field Common symbols H SI unit Ampere-turn per metre Other units Oersted Ampere-turn per inch In SI base units A m-1 Dimension L-1I

While **B** creates magnetic forces and torques on objects and induces currents in conducting wires, it is not always easy to calculate. For this reason, it is useful to define a magnetic **H** field[note 4], also known as **magnetic field strength**,[13] such that[14]:

**Definition of the H field** *([vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units))*

H ≡ 1 μ 0 B − M , {\displaystyle \mathbf {H} \equiv {\frac {1}{\mu _{0}}}\mathbf {B} -\mathbf {M} ,}

where μ 0 {\displaystyle \mu _{0}} is the [vacuum permeability](/source/Vacuum_permeability), and **M** is the [magnetization vector](/source/Magnetization) which represents how magnetized a given region of material is and is defined [below](#Magnetization). In a vacuum, **B**= μ0**H** making them equivalent to each other. Inside a material they are different (see [H and B inside and outside magnetic materials](#H-field_and_magnetic_materials)).

Defined this way, **H** can in many circumstance[note 5] be treated as if it is only due to electrical currents with corrections accounting for **H** due to nearby magnetic material.[note 6] In any case, **B** still needs to be calculated from **H** if forces, torques, induced currents, or energy changes need to be calculated.

The SI unit of **H** is the [ampere](/source/Ampere) per metre (A/m)[15] and the Gaussian unit is the [oersted](/source/Oersted) (Oe).[10]

## Measurement and visualization

### Magnetometers

A uniaxial fluxgate magnetometer

Main article: [Magnetometer](/source/Magnetometer)

Instruments used to measure the local magnetic **B**-field are known as a [magnetometers](/source/Magnetometer). Important classes of magnetometers include [induction magnetometers](/source/Search_coil) (or search-coil magnetometers) which measure only varying magnetic fields, [rotating coil magnetometers](/source/Magnetometer#Rotating_coil_magnetometer), [Hall effect](/source/Hall_effect) magnetometers, [NMR magnetometers](/source/Proton_magnetometer), [SQUID magnetometers](/source/SQUID), and [fluxgate magnetometers](/source/Magnetometer#Fluxgate_magnetometer). The magnetic fields of distant [astronomical objects](/source/Astronomical_object) are measured through their effects on local charged particles. For instance, electrons spiraling around a field line produce [synchrotron radiation](/source/Synchrotron_radiation) that is detectable in [radio waves](/source/Radio_waves). The finest precision for a magnetic field measurement was attained by [Gravity Probe B](/source/Gravity_Probe_B) at 5 aT (5×10−18 T).[16]

The **H**-field cannot be directly measured but can be inferred from the currents that create it.

### Magnetic field lines

Visualizing magnetic fields

Left: the direction of magnetic [field lines](/source/Field_line) represented by [iron filings](/source/Iron_filings) sprinkled on paper placed above a bar magnet.

Right: [compass](/source/Compass) needles point in the direction of the local magnetic field, towards a magnet's south pole and away from its north pole.

Main article: [Field line](/source/Field_line)

Magnetic field can be visualized by a set of *magnetic field lines*, that follow the direction of the field at each point. The direction of the magnetic field at any point is parallel to the direction of nearby field lines, and the local density of field lines can be made proportional to its strength. Magnetic field lines are like [streamlines](/source/Streamlines%2C_streaklines%2C_and_pathlines) in [fluid flow](/source/Fluid_dynamics), in that they represent a continuous distribution, and a different resolution would show more or fewer lines.

Magnetic field lines have the following properties:[17]

- The direction of the magnetic field is tangent to the field line at any point. A small compass points in the direction of the field line.

- The strength of the field is proportional to the closeness of the lines.

- Magnetic field lines never cross.

- Magnetic field lines form closed loops enclosing electrical currents.

- Magnetic field lines are directed from the north pole to the south pole.

An advantage of using magnetic field lines as a representation is that many laws of magnetism (and electromagnetism) can be stated completely and concisely using simple concepts such as the "number" of field lines through a surface. These concepts can then be "translated" to their mathematical form. For example, the number of field lines through a given surface is the [surface integral](/source/Surface_integral) of the magnetic field.[18]

### Different unit systems

This article uses almost entirely the SI unit system. But other unit systems, most importantly the [Gaussian](/source/Gaussian_units) unit system (which is the most used system of [cgs](/source/Cgs) units for electromagnetism), are still being used in some disciplines, countries, and textbooks. It is important to note that the equations for each unit system can and often are different for different unit system. This article, unless stated otherwise, uses equations that are only valid for the SI unit system.

## Force on moving charges and current

Moving electric charges including electrical currents experience a force due to magnetic **B** fields.

### Magnetic force on a charged particle

A charged particle moving with velocity **v** in a magnetic field **B** feels a magnetic force **F** whose direction is determined by the [right hand rule](/source/Right_hand_rule).

Main article: [Lorentz force](/source/Lorentz_force)

A [charged particle](/source/Charged_particle) moving in a **B**-field experiences a *sideways* force that is proportional to the strength of the magnetic field, the component of the velocity that is perpendicular to the magnetic field and the charge of the particle. This force is known as the *Lorentz force*, and is given by:[6][7]

**Lorentz force law**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = q E + q ( v × B ) , {\displaystyle \mathbf {F} =q\mathbf {E} +q(\mathbf {v} \times \mathbf {B} ),}

where **F** is the [force](/source/Force), *q* is the [electric charge](/source/Electric_charge) of the particle, **v** is the instantaneous [velocity](/source/Velocity) of the particle, and **B** is the magnetic field (in [teslas](/source/Tesla_(unit))). The direction of force on the charge can be determined by the [right-hand rule](/source/Right-hand_rule) (see the figure).

The Lorentz force is always perpendicular to both the velocity of the particle and the magnetic field that created it. When a charged particle moves in a static magnetic field, it traces a helical path in which the helix axis is parallel to the magnetic field, and in which the speed of the particle remains constant.

### Force on current-carrying wire

Main article: [Laplace force](/source/Laplace_force)

When a wire carrying a steady [electric current](/source/Electric_current) is placed in an external magnetic field, each of the moving charges in the wire experience the Lorentz force. Together, these forces produce a net macroscopic force on the wire. This force (on a macroscopic current) is often referred to as the *Laplace force*.

For a straight, stationary wire in a uniform magnetic field, this force is given by:[19]

**Magnetic force on straight line current**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = I ℓ × B , {\displaystyle \mathbf {F} =I{\boldsymbol {\ell }}\times \mathbf {B} \,,}

where I is the current and **ℓ** is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of the current.

If the wire is not straight or the magnetic field is non-uniform, the total force can be computed by applying the formula to each [infinitesimal](/source/Infinitesimal) segment of wire d ℓ {\displaystyle \mathrm {d} {\boldsymbol {\ell }}} , then adding up all these forces by [integration](/source/Integration_(calculus)). In this case, the net force on a stationary wire carrying a steady current is[20]

**Magnetic force on current of arbitrary shape**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = I ∫ ( d ℓ × B ) . {\displaystyle \mathbf {F} =I\int (\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {B} )\,.}

This force creates an attractive/repulsive force between 2 parallel wires as the current through each produces a magnetic field that pushes/pulls on the other. Too, a loop of current in a magnetic field will experience a torque due to the different direction of the force on different sides of the loop as describe in the next section.

### Net force and torque on current loops

A rectangular loop of current in a magnetic field,**B**, experiences a torque (around the dashed line).

A magnetic field acting on a current carrying loop produces both a torque and a net force (if the magnetic field is non-uniform).[21] This effect is important for driving certain types of motors and in modeling forces and torques on atoms.

Calculating the torque on a rectangular loop is straightforward. The diagram to the right shows a rectangular loop of current in a uniform magnetic **B** field (with a direction indicated by the green arrows). For simplicity the loop is aligned so that it is along the direction of the magnetic field. The magnetic force on opposite sides of the loop are equal and opposite producing no net force on the loop. The forces on the short sides (here shown as violet arrows), though, produce a net torque equal to the product of the force and the perpendicular distance between them. Denoting the short side length as b, the magnitude of that force is F = IBb using the equation for the magnetic force on a straight wire given in the previous section. The magnitude of the net torque (along dashed axis) is therefore N = IabB. Using the fact that the area A = ab and generalizing for all angles gives[22]

**Torque on a current loop**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

N = I A × B . {\displaystyle \mathbf {N} =I\mathbf {A} \times \mathbf {B} \,.}

Here the direction of the area **A** is the normal to the area as determined by the [right hand grip rule](/source/Right_hand_grip_rule) of the current loop. While derived for a rectangular loop this equation is valid for a flat loop of any shape and orientation.[23] As described above, there is no net force on a loop in a uniform magnetic field. However, non-uniform magnetic fields do produce a net force. This net force tends to pull the object in direction of the stronger magnetic field.

### Net force and torque on a magnetic dipole

Main article: [magnetic dipole](/source/Magnetic_dipole)

Since the net force on a loop is proportional to the current of the loop times it area, it is natural to define a quantity **m** called the **magnetic dipole moment** such that[24]

**Definition of magnetic dipole moment, m**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

m = I A . {\displaystyle \mathbf {m} =I\mathbf {A} \,.}

For a sufficiently small current loop, the details of the current loop such as it shape, area, orientation, and current around the loop are all hidden in **m** and otherwise do not matter. Such loops are called [magnetic dipoles](/source/Magnetic_dipole). All magnetic dipoles with the same dipole moment **m** are affected the same way.

Applying the Lorentz force to a (sufficiently small) current loop of arbitrary shape produces a torque **N** on the magnetic dipole of:[25]

**Torque on a magnetic dipole**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

N = m × B {\displaystyle \mathbf {N} =\mathbf {m} \times \mathbf {B} }

and a force **F** on the magnetic dipole of[26]

**Force on a magnetic dipole**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = ∇ ( m ⋅ B ) , {\displaystyle \mathbf {F} =\nabla (\mathbf {m} \cdot \mathbf {B} ),}

where ∇ {\displaystyle \nabla } represents the [gradient](/source/Gradient). This force tends to push the magnetic dipole into the direction of increasing **B**.

## Magnetic field due to electrical currents

Main articles: [Electromagnet](/source/Electromagnet), [Biot–Savart law](/source/Biot%E2%80%93Savart_law), and [Ampère's law](/source/Amp%C3%A8re's_circuital_law)

All moving charged particles produce magnetic fields. Moving [point](/source/Point_particle) charges, such as [electrons](/source/Electron), produce [complicated but well known magnetic fields](/source/Jefimenko's_equations) that depend on the charge, velocity, and acceleration of the particles.[27] These equations become much simpler when the moving charges form a steady state electrical current, the study of which is called [magnetostatics](/source/Magnetostatics).

### Magnetic field of a long straight wire

[Right hand grip rule](/source/Right_hand_grip_rule): a current flowing in the direction of the white arrow produces a magnetic field shown by the red arrows.

In general, magnetic field lines form [concentric](/source/Concentric) circles around a current-carrying wire. The direction of such a magnetic field can be determined by using the "[right-hand grip rule](/source/Right-hand_grip_rule)" (see figure at right). The strength of the magnetic field decreases with distance from the wire. (For an infinite length wire the strength is inversely proportional to the distance.) The magnetic field of a steady current I through a sufficiently long straight wire is:[28]

**Magnetic field of infinite wire**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B = μ 0 2 π r I ϕ ^ , H = I 2 π r ϕ ^ , {\displaystyle {\begin{aligned}\mathbf {B} ={\frac {\mu _{0}}{2\pi r}}I\,{\hat {\phi }},\\\mathbf {H} ={\frac {I}{2\pi r}}\,{\hat {\phi }},\end{aligned}}}

where *r* is the perpendicular distance to the wire. The direction ϕ ^ {\displaystyle {\hat {\phi }}} of the magnetic field is tangent to a circle perpendicular to the wire according to the right hand rule.[29]

### Magnetic field of an arbitrarily shaped thin wire

The direction of the magnetic flux density d**B** due to a current of a tiny element Idl varies with location **r**.

Main article: [Biot-Savart law](/source/Biot-Savart_law)

More specifically, the magnetic field generated by a steady current I (a constant flow of electric charges, in which charge neither accumulates nor is depleted at any point)[note 7] is described by the *[Biot–Savart law](/source/Biot%E2%80%93Savart_law)*:[31][32]

**Biot-Savart Law**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B = μ 0 I 4 π ∫ w i r e d ℓ × r ^ r 2 , H = I 4 π ∫ w i r e d ℓ × r ^ r 2 , {\displaystyle {\begin{aligned}\mathbf {B} ={\frac {\mu _{0}I}{4\pi }}\int _{\mathrm {wire} }{\frac {\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {\hat {r}} }{r^{2}}},\\\mathbf {H} ={\frac {I}{4\pi }}\int _{\mathrm {wire} }{\frac {\mathrm {d} {\boldsymbol {\ell }}\times \mathbf {\hat {r}} }{r^{2}}},\end{aligned}}}

where the integral sums over the wire length where vector d**ℓ** is the vector [line element](/source/Line_element) with direction in the same sense as the current *I*, *μ*0 is the [magnetic constant](/source/Magnetic_constant), *r* is the distance between the location of d**ℓ** and the location where the magnetic field is calculated, and **r̂** is a unit vector in the direction of **r**.

### Magnetic field of a solenoid

A [Solenoid](/source/Solenoid) with electric current running through it behaves like a magnet.

Main article: [solenoid](/source/Solenoid)

Bending a current-carrying wire into a loop concentrates the magnetic field inside the loop while weakening it outside. Bending a wire into multiple closely spaced loops to form a coil or '[solenoid](/source/Solenoid)' enhances this effect. A device so formed around an iron [core](/source/Magnetic_core) may act as an [electromagnet](/source/Electromagnet), generating a strong, well-controlled magnetic field.

An infinitely long solenoid has a uniform magnetic field inside, and no magnetic field outside. The magnetic field only exists inside of the solenoid and is[33]

**Magnetic field of an infinite solenoid**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

H = n I , {\displaystyle \mathbf {H} =nI,}

where n is the number of turns per unit length of the solenoid and the direction of **H** is along the length of the solenoid. A finite length solenoid produces a more complicated magnetic field that can be evaluated mathematically.

For other examples of using the Biot-Savart law to calculate the magnetic fields for other common current configurations see [#Common formulæ](#Common_formulæ) below.

### Magnetic field of a flat loop of current (magnetic dipole)

Main article: [Magnetic dipole § External magnetic field produced by a magnetic dipole moment](/source/Magnetic_dipole#External_magnetic_field_produced_by_a_magnetic_dipole_moment)

See also: [Multipole expansion](/source/Multipole_expansion)

Calculating the magnetic field strength **H** a distance z directly above the center of a circular current loop of radius a.

The magnetic field of a circular current loop of radius a and carrying a current I can be calculated straightforwardly from the Biot-Savart law for locations a distance z directly above the center of the loop:[34][35]

**Magnetic field at distance z directly above a circular current loop**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B = μ 0 m 2 π ( a 2 + z 2 ) 3 / 2 , H = m 2 π ( R 2 + z 2 ) 3 / 2 , {\displaystyle {\begin{aligned}\mathbf {B} &=\mu _{0}{\frac {\mathbf {m} }{2\pi (a^{2}+z^{2})^{3/2}}}\,,\\\mathbf {H} &={\frac {\mathbf {m} }{2\pi (R^{2}+z^{2})^{3/2}}}\,,\\\end{aligned}}}

where m = I A = I ( π a 2 z ^ ) {\displaystyle \mathbf {m} =I\mathbf {A} =I(\pi a^{2}\mathbf {\hat {z}} )} is the same magnetic dipole moment used in calculating the force and torque on a loop of current in [#Net force and torque on a magnetic dipole](#Net_force_and_torque_on_a_magnetic_dipole) above. Calculating the on-axis magnetic fields of a square loop (and other flat geometries) yields similar equations that have the same equation at long distances as the circle: H = m 2 π z 3 {\displaystyle \mathbf {H} ={\frac {\mathbf {m} }{2\pi z^{3}}}} .

Calculating the magnetic field at a arbitrary location **r** (not just on-axis) from an arbitrarily shaped current loop involves advanced math.[36] But, for sufficiently long distances, the result depends only on the magnetic moment **m** of that loop and simplifies to:[37]

**Magnetic field of magnetic dipole**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

H d i p = 1 4 π [ 3 r ^ ( m ⋅ r ^ ) − m r 3 ] = B d i p μ 0 . {\displaystyle \mathbf {H} _{dip}={\frac {1}{4\pi }}\left[{\frac {3\mathbf {\hat {r}} (\mathbf {m} \cdot \mathbf {\hat {r}} )-\mathbf {m} }{r^{3}}}\right]={\frac {\mathbf {B} _{dip}}{\mu _{0}}}.}

This equation shows that at sufficiently long distances the detailed geometry of a magnet can be replaced by a single quantity, the magnetic dipole moment **m**. This equation, therefore makes a good model for the magnetic field of atoms and can be extended to describe magnetic material. Too, it has some utility in calculating the long distance force between magnets.

### Ampere's law

Main article: [Ampère's circuital law](/source/Amp%C3%A8re's_circuital_law)

A slightly more general[38][note 8] way of relating the current I {\displaystyle I} to the **B**-field is through [Ampère's law](/source/Amp%C3%A8re's_circuital_law):[39][40]

**Ampere's Law**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

∮ B ⋅ d ℓ = μ 0 I e n c , ∮ H ⋅ d ℓ = I e n c , {\displaystyle {\begin{aligned}\oint \mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}&=\mu _{0}I_{\mathrm {enc} },\\\oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}&=I_{\mathrm {enc} },\end{aligned}}}

where the [line integral](/source/Line_integral) is over any arbitrary loop and I enc {\displaystyle I_{\text{enc}}} is the current enclosed by that loop. The I enc {\displaystyle I_{\text{enc}}} is slightly different for the 2 equations in that **B** includes the difficult to calculate bound current in magnetic material while **H** does not.[note 9] Ampère's law is always valid for steady currents and can be used to easily calculate the magnetic fields of certain highly symmetric situations such as an infinite wire or an infinite solenoid.

In a modified form that accounts for time varying electric fields, Ampère's law is one of four [Maxwell's equations](/source/Maxwell's_equations) that describe electricity and magnetism.

## Force between magnets

Main article: [Force between magnets](/source/Force_between_magnets)

### Magnets

Main article: [magnet](/source/Magnet)

[Magnets](/source/Magnet) are objects that both create their own magnetic field and respond to the magnetic field of other magnets and magnetized materials. The interaction between [magnets](/source/Magnet) and their interaction with magnetic field is extremely complicated. The correct description involves describing each magnet as being made of many small volumes of magnetic material each of which creates its own magnetic field and responds to the magnetic field of the other volumes. Such models are often extremely complex. Fortunately, in many cases, it is sufficient to understand magnets as objects that have 2 equal but opposite magnetic poles: the magnetic north and south poles. Opposite poles attract with a force that increases with smaller distances while like poles repel in the same way. Such a model is called a *magnetic pole model* and it, in some cases described below, can be used to make good quantitative predictions.

Specifying the [force between magnets](/source/Force_between_magnets) is quite complicated because it depends on the strength and [orientation](/source/Orientation_(geometry)) of both magnets and their distance and direction relative to each other. The force is particularly sensitive to rotations of the magnets due to magnetic torque. The force on each magnet depends on its magnetic moment and the magnetic field[note 10] of the other. For short distances (small r) the forces can be quite strong but it decreases quite rapidly (1/r4) for large distances.

### Force between magnets at long distances (dipole–dipole interaction)

Main article: [Magnetic dipole–dipole interaction](/source/Magnetic_dipole%E2%80%93dipole_interaction)

For 2 sufficiently small magnets, such as 2 atoms far enough away from each other, the magnetic force can be represented as that of two infinitesimally small dipoles. Using vector notation, the force, **F** of a magnetic dipole **m**1 on the magnetic dipole **m**2 is:

**The magnetic dipole–dipole interaction**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = 3 μ 0 4 π r 5 [ ( m 1 ⋅ r ) m 2 + ( m 2 ⋅ r ) m 1 + ( m 1 ⋅ m 2 ) r − 5 ( m 1 ⋅ r ) ( m 2 ⋅ r ) r 2 r ] {\displaystyle \mathbf {F} ={\frac {3\mu _{0}}{4\pi r^{5}}}\left[(\mathbf {m} _{1}\cdot \mathbf {r} )\mathbf {m} _{2}+(\mathbf {m} _{2}\cdot \mathbf {r} )\mathbf {m} _{1}+(\mathbf {m} _{1}\cdot \mathbf {m} _{2})\mathbf {r} -{\frac {5(\mathbf {m} _{1}\cdot \mathbf {r} )(\mathbf {m} _{2}\cdot \mathbf {r} )}{r^{2}}}\mathbf {r} \right]}

where **r** is the distance-vector from dipole moment **m**1 to dipole moment **m**2, with *r* = ‖***r***‖. The force acting on **m**1 is in the opposite direction. The net force depends on the orientation of both dipole moments relative to each other and relative to the distance-vector between them and it decreases rapidly (proportional to 1/r4).

### Force between magnets at moderate distance (Coulomb's law for magnetism)

Main article: [Coulomb's law for magnetism](/source/Coulomb's_law_for_magnetism)

For moderate distances it is often to sufficient model the force between magnets as the ***H**-field* of one magnet pushes and pulls on *both* poles of a second magnet. If this **H**-field is the same at both poles of the second magnet then there is no net force on that magnet since the force is opposite for opposite poles. If, however, the magnetic field of the first magnet is *nonuniform* (such as the **H** near one of its poles), each pole of the second magnet sees a different field and is subject to a different force. This difference in the two forces moves the magnet in the direction of increasing magnetic field and may also cause a net torque.

If both poles are small enough to be represented as single points then they can be considered to be point magnetic charges. [Classically](/source/Classical_mechanics), the force **F** between two magnetic poles is given by:[41]

**Coulomb's law for magnetism (force between poles)**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = μ q m 1 q m 2 4 π r 2 r ^ {\displaystyle \mathbf {F} ={{\mu q_{m1}q_{m2}} \over {4\pi r^{2}}}\mathbf {\hat {r}} }

where qm1 and qm2 are the **magnetic pole strengths** of each magnet (SI unit: [ampere](/source/Ampere)-[meter](/source/Meter)), *μ* is the [permeability](/source/Permeability_(electromagnetism)) of the intervening medium, and **r** is the separation distance between the 2 poles. Note that for 2 magnets (each having 2 poles) the sum of 4 forces is needed: each of the 2 poles of one magnet exerts a separate force on each of the 2 poles of the second magnet.

The pole description is useful to practicing magneticians who design real-world magnets, but real magnets have a pole distribution more complex than a single north and south. Therefore, implementation of the pole idea is not simple. In some cases, one of the more complex formulas given below will be more useful.

### Magnetic force at small distances (pull force)

Main article: [Magnetic pull force](/source/Magnetic_pull_force)

The mechanical force between two nearby magnetized surfaces can be calculated with the following equation. The equation is valid only for cases in which the effect of fringing is negligible and the volume of the air gap is much smaller than that of the magnetized material, the force for each magnetized surface is:[42][43][44]

**Magnetic pull between nearly touching magnetic poles (magnetic pole nearly touching iron)**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

F = μ 0 H 2 A 2 = B 2 A 2 μ 0 {\displaystyle F={\frac {\mu _{0}H^{2}A}{2}}={\frac {B^{2}A}{2\mu _{0}}}}

where *A* is the surface area of the magnetic pole and μ0 is the [permeability of free space](/source/Permeability_of_free_space). This equation is also valid for the force of a magnetic pole on iron that is either almost touching or touching the magnetic pole.

### Magnetic torque on permanent magnets

Main article: [Magnetic torque](/source/Magnetic_torque)

If two like poles of two separate magnets are brought near each other, and one of the magnets is allowed to turn, it promptly rotates to align itself with the first. In this example, the magnetic field of the stationary magnet creates a *magnetic torque* on the magnet that is free to rotate. This magnetic torque **N** tends to align a magnet's poles with the magnetic field lines. A compass, therefore, turns to align itself with Earth's magnetic field.

Torque on a dipole

In the pole model of a dipole, an **H** field (to right) causes equal but opposite forces on a N pole (+*q*) and a S pole (−*q*) creating a torque.

Equivalently, a **B** field induces the same torque on a current loop with the same magnetic dipole moment.

Mathematically, the torque **N** on a small magnet is proportional both to the applied magnetic field and to the magnetic moment **m** of the magnet:[45]

**Magnetic torque**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

N = m × B {\displaystyle {\boldsymbol {N}}=\mathbf {m} \times \mathbf {B} }

where × represents the vector [cross product](/source/Cross_product). This equation includes all of the qualitative information included above. There is no torque on a magnet if **m** is in the same direction as the magnetic field, since the cross product is zero for two vectors that are in the same direction. Further, all other orientations feel a torque that twists them toward the direction of magnetic field.

## Magnetic field due to magnetized material

Main article: [Magnet](/source/Magnet)

Most materials respond to an applied magnetic field by becoming magnetized (at least temporarily) which causes them to produce their own magnetic field. Typically, the response is weak and exists only when the magnetic field is applied. There are many different types of material that respond differently to the applied magnetic field.

### Types of magnetic materials

Main article: [Magnetism](/source/Magnetism)

The term *magnet* is typically reserved for objects that produce their own persistent magnetic field even in the absence of an applied magnetic field. Only certain classes of materials can do this. Most materials, however, produce a magnetic field in response to an applied magnetic field – a phenomenon known as [magnetism](/source/Magnetism). There are several types of magnetism, and all materials exhibit at least one of them.

The overall magnetic behavior of a material can vary widely, depending on the structure of the material, particularly on its [electron configuration](/source/Electron_configuration). It can also vary with temperature, pressure, and magnetic field strength such that a given material may have more than one magnetic [phase](/source/Phase_(matter)). Several forms of magnetic behavior have been observed in different materials, including:

- [Diamagnetism](/source/Diamagnetism)[46] produces a magnetization that opposes the magnetic field.

- [Paramagnetism](/source/Paramagnetism)[46] produces a magnetization in the same direction as the applied magnetic field.

- [Ferromagnetism](/source/Ferromagnetism) and the closely related [Ferrimagnetism](/source/Ferrimagnetism) and [Antiferromagnetism](/source/Antiferromagnetism)[47][48] can produce a magnetization independent of the applied magnetic field with a complicated and often [hysteretic](/source/Hysteresis) relationship. Materials in these states can be used to make permanent magnets.

- [Superconductivity](/source/Superconductivity) (and [ferromagnetic superconductors](/source/Ferromagnetic_superconductor))[49][50] is characterized by perfect conductivity below a critical temperature and magnetic field. They also are highly magnetic and can be perfect diamagnets below a lower [critical magnetic field](/source/Critical_magnetic_field). Superconductors often have a broad range of temperatures and magnetic fields (the so-named [mixed state](/source/Type_II_superconductor#Mixed_state)) under which they exhibit a complicated and often hysteretic relationship between how the material is magnetized and the applied magnetic field.

In the case of paramagnetism and diamagnetism, the relationship between the applied magnetic field and the magnetization is often [linear](/source/Linear). However, superconductors and ferromagnets have a more complicated relation between the applied magnetic field and magnetization produced (see [magnetic hysteresis](/source/Hysteresis#Magnetic_hysteresis)). *Permanent magnets* are objects that produce their own persistent magnetic fields. They are made of [ferromagnetic](/source/Ferromagnetism)[note 11] materials, such as iron and [nickel](/source/Nickel), that have been magnetized.

### Magnetic dipole moment

The Amperian loop model

A current loop (ring) that goes into the page at the x and comes out at the dot produces a **B**-field (lines). As the radius of the current loop shrinks, the fields produced become identical to an abstract "magnetic dipole" (represented by an arrow pointing to the right).

Main articles: [magnetic dipole](/source/Magnetic_dipole) and [magnetic dipole moment](/source/Magnetic_dipole_moment)

The magnetic field of magnetized material is created at the atomic level. The proper description of this effect involves [quantum mechanics](/source/Quantum_mechanics). Fortunately, the net effect of adding up these magnetic interactions can often be calculated using much simpler models for the magnetic field created by the constituent atoms in the magnetic material. This occurs because at large enough distance (or equivalently for small enough magnets) all the magnetic properties of any magnetic object can be described by a single (vector) quantity, the [magnetic dipole moment](/source/Magnetic_dipole_moment), **m**. (See [#Magnetic field of a flat loop of current (magnetic dipole)](#Magnetic_field_of_a_flat_loop_of_current_(magnetic_dipole)) and [#Net force and torque on a magnetic dipole](#Net_force_and_torque_on_a_magnetic_dipole) above). Objects that can be modeled this way, for example atoms, are called [magnetic dipoles](/source/Magnetic_dipoles).

Magnetic dipoles, therefore, are the building blocks of magnetization. The magnetic field produced by magnetized material then is the net magnetic field of these dipoles. Too, the net force (and torque) on a magnetized material is a result of adding up the forces and torques on the individual dipoles that make up the magnetized material.

### Magnetization

Main article: [Magnetization](/source/Magnetization)

The *magnetization* vector field **M** represents how strongly a region of material is magnetized. It is defined as the net [magnetic dipole moment](/source/Magnetic_dipole_moment) per unit volume of that region.[51] The magnetization of a uniformly magnetized magnet is therefore a constant, equal to the magnetic moment **m** of the magnet divided by its volume. Since the SI unit of magnetic moment is A⋅m2, the SI unit of magnetization **M** is ampere per meter, identical to that of the **H**-field.

The magnetization **M** field of a region points in the direction of the average magnetic dipole moment in that region. Magnetization field lines, therefore, begin (inside the magnetized material) near the magnetic south pole and ends (inside the magnetized material) near the magnetic north pole. (Magnetization does not exist outside magnetized material.)

In the Amperian loop model, the magnetization is due to combining many tiny magnetic dipole loops to form a resultant current called *[bound current](/source/Bound_current)*. This bound current, then, is the source of the magnetic **B** field due to the magnet. Given the definition of the magnetic dipole, the magnetization field follows a similar law to that of Ampere's law:[52]

**Relation between M and bound current**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

∮ M ⋅ d ℓ = I b , {\displaystyle \oint \mathbf {M} \cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {b} }\,,}

where the integral is a line integral over any closed loop and *I*b is the bound current enclosed by that closed loop.

Unlike the magnetic **B** field-lines which cannot begin nor end, magnetization field lines can begin and end. Indeed they must begin and end where they intersects the boundary of the magnetized material (at magnetic poles) because the magnetization field only exists inside of a material. This is analogous to [electric field](/source/Electric_field)-lines which begin and end at electrical charges. It is therefore possible to define a 'magnetic charge' qm such that for a given region the net 'magnetic charge' is:[53]

**Relation between M and fictitious magnetic charge**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

∮ S μ 0 M ⋅ d A = − q m , {\displaystyle \oint _{S}\mu _{0}\mathbf {M} \cdot \mathrm {d} \mathbf {A} =-q_{\mathrm {m} }\,,}

where the integral is a closed surface integral over the closed surface *S* and *q*M is the "magnetic charge" (in units of [magnetic flux](/source/Magnetic_flux)) enclosed by *S*. (A closed surface completely surrounds a region with no holes to let any field lines escape.) The negative sign occurs because the magnetization field moves from south to north. It is important to note that no such magnetic charge exists; rather it is a convenient analogy that allows the use of much of the machinery developed for electrostatics with electric charge to be applied to magnetization with its fictitious magnetic charge. For example the net magnetic charge of a pole is defined as a **magnetic pole strength** qm.

### Relation between B, H, and M

Comparison of **B**, **H** and **M** inside and outside a cylindrical bar magnet.

See also: [Demagnetizing field](/source/Demagnetizing_field)

Using the above definition of **M** it is now possible to define the magnetic **H** field[54]

**Definition of H**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

H ≡ B μ 0 − M . {\displaystyle \mathbf {H} \ \equiv \ {\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} .}

In terms of the H-field, Ampere's law is:[55]

**Relation between H and free current**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

∮ H ⋅ d ℓ = ∮ ( B μ 0 − M ) ⋅ d ℓ = I t o t − I b = I f , {\displaystyle \oint \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}=\oint \left({\frac {\mathbf {B} }{\mu _{0}}}-\mathbf {M} \right)\cdot \mathrm {d} {\boldsymbol {\ell }}=I_{\mathrm {tot} }-I_{\mathrm {b} }=I_{\mathrm {f} },}

where If represents the 'free current' enclosed by the loop so that the line integral of **H** does not depend at all on the bound currents.[56]

Similarly, a [surface integral](/source/Surface_integral) of **H** over any [closed surface](/source/Closed_surface) is independent of the free currents and picks out the "magnetic charges" within that closed surface:

**Relation between H and fictitious magnetic charge**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

∮ S μ 0 H ⋅ d A = ∮ S ( B − μ 0 M ) ⋅ d A = 0 − ( − q M ) = q m , {\displaystyle \oint _{S}\mu _{0}\mathbf {H} \cdot \mathrm {d} \mathbf {A} =\oint _{S}(\mathbf {B} -\mu _{0}\mathbf {M} )\cdot \mathrm {d} \mathbf {A} =0-(-q_{\mathrm {M} })=q_{\mathrm {m} }\,,}

which does not depend on the free currents.

The **H**-field, therefore, can be separated into two[note 12] independent parts: H = H 0 + H d {\displaystyle \mathbf {H} =\mathbf {H} _{0}+\mathbf {H} _{\mathrm {d} }} , where **H**0 is the applied magnetic field due only to the free currents and **H**d is the [demagnetizing field](/source/Demagnetizing_field) due only to the bound currents which can equivalently be expressed in terms of the fictitious magnetic charge qm. The magnetic **H**-field, therefore, re-factors the bound current in terms of "magnetic charges". The **H** field lines loop only around "free current" and, unlike the magnetic **B** field, begins and ends near magnetic poles as well.

### Constitutive relation between B and H

Main articles: [permeability (electromagnetism)](/source/Permeability_(electromagnetism)) and [magnetic susceptibility](/source/Magnetic_susceptibility)

For many materials (particularly diamagnetic and paramagnetic materials) the relationship between **B** and **H** is linear:[57]

**Constitutive relation between B and H**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B = μ H , {\displaystyle \mathbf {B} =\mu \mathbf {H} ,}

where *μ* is a material dependent parameter called the [permeability](/source/Permeability_(electromagnetism)). In some cases the permeability may be a second rank [tensor](/source/Tensor) so that **H** may not point in the same direction as **B**. These relations between **B** and **H** are examples of [constitutive equations](/source/Constitutive_equation).

### Boundary conditions for B and H

Main article: [Interface conditions for electromagnetic fields § Interface conditions for magnetic field vectors](/source/Interface_conditions_for_electromagnetic_fields#Interface_conditions_for_magnetic_field_vectors)

In many real world applications such as small magnetic object inside of an extended applied magnetic field, the constitutive relation is not sufficient even if the material is linear. This is because the **H**-field that the material experiences is not the same as the **H** applied. In such cases, the magnetic field can still be calculated but care must be taken to distinguish the change of the magnetic field across the boundary of the magnetic object. These relations in the most simplified form (in terms of **H** only in a linear material and without and free current) are:[58][59]

**Boundary Conditions for H (no free current)**  (*scalar form, [SI units](/source/International_System_of_Units)*)

H 1 t = H 2 t , μ 1 H 1 N = μ 2 H 2 n , {\displaystyle {\begin{aligned}H_{1t}&=H_{2t}\,,\\\mu _{1}H_{1N}&=\mu _{2}H_{2n}\,,\end{aligned}}}

where the subscript t represents the tangential component of **H** and n represents its normal component.

## Electrodynamics

Main article: [electrodynamics](/source/Electrodynamics)

For time varying magnetic fields (and more generally changing electrical currents or accelerating electrical charges), the magnetic and electric fields become linked such that a change in one induces the other. Together, the electric and magnetic fields form an [electromagnetic field](/source/Electromagnetic_field). The study of how the electric and magnetic fields interact in this way is called [electrodynamics](/source/Electrodynamics) and includes many phenomenon that are important in physics and electrical engineering. It underlies transformers, and the generation and transmission of electrical power through wires and through space in the form of [electromagnetic radiation](/source/Electromagnetic_radiation) of which [light](/source/Light) is one form. Too, it allow magnetic fields to store and transmit energy.

### Magnetic flux rule

Main article: [Faraday's law of induction](/source/Faraday's_law_of_induction)

A time varying magnetic field through a loop of wire induces a current (more properly an [EMF](/source/Electromotive_force)) through that loop. This is known as [electromagnetic induction](/source/Electromagnetic_induction) and is important for many electronic devices such as [inductors](/source/Inductor), [transformers](/source/Transformer), and [electrical generators](/source/Electrical_generator). The equation governing this is known as the **flux rule** or [Faraday's law of induction](/source/Faraday's_law_of_induction):[60]

**Magnetic flux rule (Faraday's law of induction)**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

E = − d Φ d t , {\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi }{\mathrm {d} t}}\,,}

where E {\displaystyle {\mathcal {E}}} is the electromotive force (or *EMF*, the [voltage](/source/Voltage) generated around a closed loop) and Φ is the [magnetic flux](/source/Magnetic_flux)—the product of the area times the magnetic field [normal](/source/Tangential_and_normal_components) to that area. (This definition of magnetic flux is why **B** is often referred to as *magnetic flux density*.)[61] The negative sign represents the fact that any current generated by a changing magnetic field in a coil produces a magnetic field that *opposes* the *change* in the magnetic field that induced it. This phenomenon is known as [Lenz's law](/source/Lenz's_law).

### Stored energy

Main article: [Magnetic energy](/source/Magnetic_energy)

See also: [Magnetic hysteresis](/source/Magnetic_hysteresis)

Energy is needed to generate a magnetic field both to work against the electric field that a changing magnetic field creates and to change the magnetization of any material within the magnetic field. The energy density of just creating the field at a given region is:[62]

**Magnetic energy density in vacuum**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

u m a g = B ⋅ B 2 μ 0 . {\displaystyle u_{mag}={\frac {\mathbf {B} \cdot \mathbf {B} }{2\mu _{0}}}\,.}

For non-dispersive materials, the energy used to magnetize the material is released when the magnetic field is destroyed so that the energy can be modeled as being stored in the magnetic field. If the non-dispersive material is also linear (such that **B** = *μ***H** where *μ* is frequency-independent), then the total energy density stored in the magnetic field and in magnetizing the material at a location is:[63]

**Magnetic energy density in linear material**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

u m a g = B ⋅ H 2 = B ⋅ B 2 μ = μ H ⋅ H 2 . {\displaystyle u_{mag}={\frac {\mathbf {B} \cdot \mathbf {H} }{2}}={\frac {\mathbf {B} \cdot \mathbf {B} }{2\mu }}={\frac {\mu \mathbf {H} \cdot \mathbf {H} }{2}}\,.}

The above equation cannot be used for nonlinear materials, though. In general, the incremental amount of work per unit volume *δW* needed to cause a small change of magnetic field *δ***B** is:[64]

**Differential work done (per unit volume) in creating a magnetic field in the presence of a material**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

δ W = H ⋅ δ B . {\displaystyle \delta W=\mathbf {H} \cdot \delta \mathbf {B} \,.}

Once the relationship between **H** and **B** is known this equation is used to determine the work needed to reach a given magnetic state. For [hysteretic materials](/source/Hysteresis) such as ferromagnets and superconductors, the work needed also depends on how the magnetic field is created. For linear non-dispersive materials, though, the general equation leads directly to the simpler energy density equation given above.

### Poynting vector

Main article: [Poynting vector](/source/Poynting_vector)

Illustration of electromagnetic power flow inside a [coaxial cable](/source/Coaxial_cable) according to the Poynting vector **S**, calculated using the electric field **E** (due to the voltage *V*) and the magnetic field **H** (due to current I).

Magnetic field, together with the electric field, transmit electrical power. The amount of electrical power (per unit area) transmitted this way is called the **poynting vector**, **S**, which depends on the magnetic field as the [cross product](/source/Cross_product):[65][66]

**The Poynting vector (power per unit area passing a given point)**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

S = E × H , {\displaystyle \mathbf {S} =\mathbf {E} \times \mathbf {H} \,,}

where **E** is the [electric field](/source/Electric_field). Note that this power includes both the power transmitted by the electric and magnetic fields and the energy absorbed and emitted by magnetizing and polarizing the material. Too, this equation only works for linear non-dispersive materials. This equations is also valid in a vacuum where **H** = **B**/μ0.

The time average of the poynting vector is known as [irradiance](/source/Irradiance) and is an important quantity in [optics](/source/Optics) that describes how intense light is at a given point.

### Maxwell's equations

Main article: [Maxwell's equations](/source/Maxwell's_equations)

It is sometimes useful to calculate the magnetic field for a given set of time varying charges and currents, without having to use the [complicated equations](/source/Jefimenko's_equations) used to directly calculate it. An example of this is calculating the magnetic field of a light wave as it reflects and refracts at a surface. In such cases [Maxwell's equations](/source/Maxwell's_equations) are used to solve for both the magnetic and electric fields. (In electrodynamics the electric and magnetic fields are coupled.)

Maxwell's equations are a powerful set of [differential equations](/source/Differential_equation) that allows the calculation of the magnetic and electric fields for simple (and complex using computers and [Finite Element Analysis](/source/Finite_Element_Analysis)) geometries. Maxwell's Equations together with the Lorentz force law form a complete description of [classical electrodynamics](/source/Classical_electromagnetism) including both electricity and magnetism.

Maxwell's equations takes advantage of the fact that all vector fields (such as the electric and magnetic fields) can be expressed in terms of 2 types of *sources* and an appropriate set of boundary conditions.[note 13] The first type of *source* (an outflow source) causes the vector field to flow out (or in for a sink) to a given point. The second (or circulation) source causes the vector field to rotate around a given point (forming [vortices](/source/Vortex)). Both of these sources have well defined definitions and can be calculated from the vector field they create using a well-understood vector operator.

The [divergence](/source/Divergence) of a vector field **A**, **∇** · **A** is defined such that applying the divergence operator to a given vector field will yield the outflow sources. The [curl](/source/Curl_(mathematics)) is defined such that **∇** × **A** yields the circulation source. An example of the power of these vector operators is: since it is an experimental fact that magnetic charges do not exist (and therefore there are no source nor sinks of **B**) the divergence of **B** must be zero, **∇** · **B** = 0, which is one of Maxwell's equations.

Maxwell's equation has 2 major versions: a microscopic version which necessitates knowing all of the charges and currents (including the complex ones at the atomic level) and the macroscopic version which depends only on the know 'free' charge and 'free' currents. Here the term 'free' means any charge or current that is directly controlled by the experiment and does not include the atomic level 'bound' charges and currents in a material which happen as a response to the electric and magnetic fields present in that material.

Maxwell's macroscopic equations are written as:

**Maxwell's equations in matter**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

∇ ⋅ D = ρ f ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × H = J f + ∂ D ∂ t . {\displaystyle {\begin{aligned}\nabla \cdot \mathbf {D} \,\,\,&=\rho _{f}\\\nabla \cdot \mathbf {B} \,\,\,&=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {H} &=\mathbf {J} _{f}+{\frac {\partial \mathbf {D} }{\partial t}}\,.\end{aligned}}}

In these equations, D {\displaystyle \mathbf {D} } is the [electric displacement field](/source/Electric_displacement_field), E {\displaystyle \mathbf {E} } the [electric field](/source/Electric_field), ρ f {\displaystyle \rho _{f}} the free [electric charge density](/source/Electric_charge_density), and J f {\displaystyle \mathbf {J} _{f}} the free [current density](/source/Current_density).

The first of Maxwell's equations is known as [Gauss' Law](/source/Gauss'_Law) but does not involve magnetic field so does not warrant further discussion here. The second equation is [Gauss' law for magnetism](/source/Gauss'_law_for_magnetism) which reflects the non-existence of magnetic charge and allows **B** to be determined as the curl of a vector potential **A**. The third equation is [Faraday's law of induction](/source/Faraday's_law_of_induction). And, the fourth equation is [Ampère's law](/source/Amp%C3%A8re's_circuital_law) with Maxwell's correction.

## Advanced formulations

### Magnetic vector potential

Main article: [Magnetic vector potential](/source/Magnetic_vector_potential)

In deriving advanced equations and in advanced topics such as [quantum mechanics](/source/Quantum_mechanics) and [relativity](/source/Theory_of_relativity), it is often easier to work with a *potential formulation* of electrodynamics rather than in terms of the electric and magnetic fields. In this representation, the *[magnetic vector potential](/source/Magnetic_vector_potential)* **A**, and the [electric scalar potential](/source/Electric_potential) *φ*, are defined such that:[67]

**Definition of the vector A and scalar φ potentials**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B = ∇ × A , E = − ∇ φ − ∂ A ∂ t . {\displaystyle {\begin{aligned}\mathbf {B} &=\nabla \times \mathbf {A} ,\\\mathbf {E} &=-\nabla \varphi -{\frac {\partial \mathbf {A} }{\partial t}}.\end{aligned}}}

The vector potential, ***A*** given by this form may be interpreted as a *generalized potential [momentum](/source/Momentum) per unit charge*[68] just as *φ* is interpreted as a *generalized [potential energy](/source/Potential_energy) per unit charge*. There are multiple choices one can make for the potential fields that satisfy the above condition. However, the choice of potentials is represented by its respective [gauge condition](/source/Gauge_fixing).

Maxwell's equations when expressed in terms of the potentials can be cast into a form[69] that explicitly agrees with [special relativity](/source/Special_relativity).[70] Together, **A** and *φ* form the [four-potential](/source/Four-potential). Using the four potential instead of electric and magnetic fields is much simpler—and it can be easily adapted to work with quantum mechanics.

### Magnetic and electric fields are different aspects of the same phenomenon

Main article: [Relativistic electromagnetism](/source/Relativistic_electromagnetism)

Magnetic field is inherently a [relativistic phenomena](/source/Special_relativity). More specifically, both electric and magnetic fields are the same phenomenon as seen in different [reference frames](/source/Frame_of_reference#Observational_frames_of_reference): An electric force perceived by one observer may be perceived by another (in a different frame of reference) as a magnetic force, or a mixture of electric and magnetic forces. (Here different reference frames means one reference frame is moving relative to the other.) For relativistic phenomena, a [lorentz transformation](/source/Lorentz_transformation) must be used to move (or transform) from one reference system to another.

It is a straightforward task[71] to show how the electric and magnetic fields transform from one reference frame to another. The [transformation rules](/source/Lorentz_transformation#Transformation_of_the_electromagnetic_field), however are quite messy. One simple example is to examine how [Coulomb's Law](/source/Coulomb's_law) (which is a pure electric field of a charged particle in it own rest frame) transforms to a moving reference frame. A point in the moving reference frame will experience a magnetic field of:[72]: 29–42

**Magnetic (and electric) field of a uniformly moving point charge**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B = q 4 π ε 0 r 3 1 − β 2 ( 1 − β 2 sin 2 ⁡ θ ) 3 / 2 v × r c 2 = v × E c 2 , {\displaystyle \mathbf {B} ={\frac {q}{4\pi \varepsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}\,,}

where q {\displaystyle q} is the charge of the point source, ε 0 {\displaystyle \varepsilon _{0}} is the [vacuum permittivity](/source/Vacuum_permittivity), r {\displaystyle \mathbf {r} } is the position vector from the point source to the point in space, v {\displaystyle \mathbf {v} } is the velocity vector of the charged particle, β {\displaystyle \beta } is the ratio of speed of the charged particle divided by the speed of light and θ {\displaystyle \theta } is the angle between r {\displaystyle \mathbf {r} } and v {\displaystyle \mathbf {v} } .

Formally, special relativity combines the electric and magnetic fields into a rank-2 [tensor](/source/Tensor), called the *[electromagnetic tensor](/source/Electromagnetic_tensor)*. Changing reference frames *mixes* these components. This is analogous to the way that special relativity *mixes* space and time into [spacetime](/source/Spacetime), and mass, momentum, and energy into [four-momentum](/source/Four-momentum).[73] Similarly, the [energy stored in a magnetic field](/source/Magnetic_energy) is mixed with the energy stored in an electric field in the [electromagnetic stress–energy tensor](/source/Electromagnetic_stress%E2%80%93energy_tensor).

### Magnetic field of arbitrarily moving point charge

Main article: [Liénard–Wiechert potential](/source/Li%C3%A9nard%E2%80%93Wiechert_potential)

The solution of maxwell's equations for electric and magnetic field of a point charge is expressed in terms of [retarded time](/source/Retarded_time) or the time at which the particle in the past causes the field at the point, given that the influence travels across space at the speed of light. The [retarded time](/source/Retarded_time) for a point particle is given as solution of:[74]

**Definition of retarded time (which enforces causality)**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

t r = t − | r − r s ( t r ) | c , {\displaystyle t_{r}=\mathbf {t} -{\frac {\left|\mathbf {r} -\mathbf {r} _{s}(t_{r})\right|}{c}}\,,}

where the retarded time t r {\textstyle t_{r}} is the time at which the source's contribution of the field originated, r s ( t ) {\textstyle r_{s}(t)} is the position vector of the particle as function of time, r {\textstyle \mathbf {r} } is the point in space, t {\textstyle \mathbf {t} } is the time at which fields are measured and c {\textstyle c} is the speed of light. Any arbitrary motion of point charge causes electric and magnetic fields as follows:[75]

**Magnetic (and electric) field of an arbitrarily moving point charge**  (*[vector](/source/Euclidean_vector) form, [SI units](/source/International_System_of_Units)*)

B ( r , t ) = μ 0 4 π [ q c ( β s × n s ) γ 2 ( 1 − n s ⋅ β s ) 3 | r − r s | 2 + q n s × ( n s × ( ( n s − β s ) × β s ˙ ) ) ( 1 − n s ⋅ β s ) 3 | r − r s | ] t = t r = n s ( t r ) c × E ( r , t , ) {\displaystyle {\begin{aligned}\mathbf {B} (\mathbf {r} ,\mathbf {t} )&={\frac {\mu _{0}}{4\pi }}\left[{\frac {qc({\boldsymbol {\beta }}_{s}\times \mathbf {n} _{s})}{\gamma ^{2}{\left(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s}\right)}^{3}{\left|\mathbf {r} -\mathbf {r} _{s}\right|}^{2}}}+{\frac {q\mathbf {n} _{s}\times \left(\mathbf {n} _{s}\times \left(\left(\mathbf {n} _{s}-{\boldsymbol {\beta }}_{s}\right)\times {\dot {{\boldsymbol {\beta }}_{s}}}\right)\right)}{{\left(1-\mathbf {n} _{s}\cdot {\boldsymbol {\beta }}_{s}\right)}^{3}\left|\mathbf {r} -\mathbf {r} _{s}\right|}}\right]_{t=t_{r}}\\[1ex]&={\frac {\mathbf {n} _{s}(t_{r})}{c}}\times \mathbf {E} (\mathbf {r} ,\mathbf {t} \,,)\end{aligned}}}

where q is the charge of the point source, n s {\displaystyle n_{s}} is a unit vector pointing from charged particle to the point in space, β s ( t ) {\displaystyle {\boldsymbol {\beta }}_{s}(t)} is the velocity of the particle divided by the speed of light and γ ( t ) {\displaystyle \gamma (t)} is the corresponding [Lorentz factor](/source/Lorentz_factor).

### Quantum electrodynamics

See also: [Standard Model](/source/Standard_Model) and [quantum electrodynamics](/source/Quantum_electrodynamics)

The classical electromagnetic field incorporated into quantum mechanics forms what is known as the semi-classical theory of radiation. However, it is not able to make experimentally observed predictions such as [spontaneous emission process](/source/Spontaneous_emission) or [Lamb shift](/source/Lamb_shift) implying the need for quantization of fields. In modern physics, the electromagnetic field is understood to be not a *[classical](/source/Classical_physics)* [field](/source/Field_(physics)), but rather a [quantum field](/source/Quantum_field); it is represented not as a vector of three [numbers](/source/Real_number) at each point, but as a vector of three [quantum operators](/source/Operator_(physics)) at each point. The most accurate modern description of the electromagnetic interaction (and much else) is *quantum electrodynamics* (QED),[76] which is incorporated into a more complete theory known as the *Standard Model of particle physics*.

In QED, the magnitude of the electromagnetic interactions between charged particles (and their [antiparticles](/source/Antiparticle)) is computed using [perturbation theory](/source/Perturbation_theory_(quantum_mechanics)). These rather complex formulas produce a remarkable pictorial representation as [Feynman diagrams](/source/Feynman_diagram) in which [virtual photons](/source/Virtual_photon) are exchanged.

Predictions of QED agree with experiments to an extremely high degree of accuracy: currently about 10−12 (and limited by experimental errors); for details see [precision tests of QED](/source/Precision_tests_of_QED). This makes QED one of the most accurate physical theories constructed thus far.

All equations in this article are in the [classical approximation](/source/Classical_electromagnetism), which is less accurate than the quantum description mentioned here. However, under most everyday circumstances, the difference between the two theories is negligible.

## Applications

### Uses in geology

Main article: [Geophysics § Magnetism](/source/Geophysics#Magnetism)

#### Earth's magnetic field

A sketch of Earth's magnetic field representing the source of the field as a magnet. The south pole of the magnetic field is near the geographic north pole of the Earth.

Main article: [Earth's magnetic field](/source/Earth's_magnetic_field)

The Earth's magnetic field is produced by [convection](/source/Convection) of a liquid iron alloy in the [outer core](/source/Outer_core). In a [dynamo process](/source/Dynamo_theory), the movements drive a feedback process in which electric currents create electric and magnetic fields that in turn act on the currents.[77]

The field at the surface of the Earth is approximately the same as if a giant bar magnet were positioned at the center of the Earth and tilted at an angle of about 11° off the rotational axis of the Earth (see the figure).[78] The north pole of a magnetic compass needle points roughly north, toward the [North Magnetic Pole](/source/North_Magnetic_Pole). However, because a magnetic pole is attracted to its opposite, the North Magnetic Pole is actually the south pole of the geomagnetic field. This confusion in terminology arises because the pole of a magnet is defined by the geographical direction it points.[79]

Earth's magnetic field is not constant—the strength of the field and the location of its poles vary.[80] Moreover, the poles periodically reverse their orientation in a process called [geomagnetic reversal](/source/Geomagnetic_reversal). The [most recent reversal](/source/Brunhes%E2%80%93Matuyama_reversal) occurred 780,000 years ago.[81]

#### Magnetic surveys

Magnetic gradiometer map of Prehistoric fire-hearths

Main articles: [Aeromagnetic survey](/source/Aeromagnetic_survey) and [Magnetic survey (archaeology)](/source/Magnetic_survey_(archaeology))

**Magnetic surveying** is one of a number of methods used in [archaeological geophysics](/source/Geophysical_survey_(archaeology)). Magnetic surveys record spatial variation in the Earth's magnetic field. In [archaeology](/source/Archaeology), magnetic surveys are used to detect and map archaeological [artefacts](/source/Artefact_(archaeology)) and [features](/source/Feature_(archaeology)). Magnetic surveys are used in both terrestrial and [marine archaeology](/source/Marine_archaeology). In terrestrial archaeology, magnetic surveys are typically used for detailed mapping of archaeological features on known archaeological sites. More exceptionally, magnetometers are used for low-resolution exploratory surveys. Magnetic survey help to prove that a survey area has the potential for more detailed studies and scientific excavation. Magnetic surveys are extremely useful in the excavation and exploration of underwater archaeological sites. In maritime archaeology, these are often used to map the geology of wreck sites and determine the composition of magnetic materials found on the seafloor.

Measuring the Earths' magnetic field is a very useful tool in [mineral exploration](/source/Mineral_exploration), [oil exploration](/source/Oil_exploration), and [geological mapping](/source/Geological_mapping). To cover large areas with uniform data, aircraft such as [helicopters](/source/Helicopter), [airplanes](/source/Airplanes), and [drones](/source/Unmanned_aerial_vehicle) are employed. The amount of detail is a function of flight height and sample density, in addition to instrument sensitivity. For surveys, drones are used which helps greatly in the process. Aeromagnetic surveys are also used to perform reconnaissance mapping of [unexploded ordnance](/source/Unexploded_ordnance).

### Uses in engineering

#### Rotating magnetic fields

Main articles: [Rotating magnetic field](/source/Rotating_magnetic_field) and [Alternator](/source/Alternator)

The *rotating magnetic field* is a common design principle in the operation of [alternating-current motors](/source/Electric_motor#AC_motors). A permanent magnet in such a field rotates so as to maintain its alignment with the external field.

Magnetic torque is used to drive [electric motors](/source/Electric_motor). In one simple motor design, a magnet is fixed to a freely rotating shaft and is subjected to a magnetic field from an array of [electromagnets](/source/Electromagnet). By continuously switching the electric current through each of the electromagnets, thereby flipping the polarity of their magnetic fields, like poles are kept next to the rotor; the resultant torque is transferred to the shaft.

A rotating magnetic field can be constructed using two coils at right angles with a phase difference of 90 degrees between their AC currents. In practice, [three-phase](/source/Three-phase_electric_power) systems are used where the three currents are equal in magnitude and have a phase difference of 120 degrees. Three similar coils at mutual geometrical angles of 120 degrees create the rotating magnetic field. The ability of the three-phase system to create a rotating field, utilized in electric motors, is one of the main reasons why three-phase systems dominate the world's [electrical power](/source/Electrical_power) supply systems.

[Synchronous motors](/source/Synchronous_motor) use DC-voltage-fed rotor windings, which lets the excitation of the machine be controlled—and [induction motors](/source/Induction_motor) use short-circuited [rotors](/source/Rotor_(electric)) (instead of a magnet) following the rotating magnetic field of a multicoiled [stator](/source/Stator_(electric_machines)). The short-circuited turns of the rotor develop [eddy currents](/source/Eddy_current) induced by the rotating field of the stator, and these currents in turn produce a torque on the rotor through the Lorentz force.

The Italian physicist [Galileo Ferraris](/source/Galileo_Ferraris) and the Serbian-American [electrical engineer](/source/Electrical_engineer) [Nikola Tesla](/source/Nikola_Tesla) independently researched the use of rotating magnetic fields in electric motors. In 1888, Ferraris published his research in a paper to the *Royal Academy of Sciences* in [Turin](/source/Turin) and Tesla gained [U.S. patent 381,968](https://patents.google.com/patent/US381968) for his work.

#### Magnetic circuits

Magnetic circuit

Main article: [Magnetic circuit](/source/Magnetic_circuit)

An important use of **H** is in *magnetic circuits*. A magnetic circuit is made up of one or more closed loop paths containing a [magnetic flux](/source/Magnetic_flux). The flux is usually generated by [permanent magnets](/source/Permanent_magnet) or [electromagnets](/source/Electromagnet) and confined to the path by [magnetic cores](/source/Magnetic_core) consisting of [ferromagnetic materials](/source/Ferromagnetic_material) like iron, although there may be air gaps or other materials in the path. Magnetic circuits are employed to efficiently channel magnetic fields in many devices such as [electric motors](/source/Electric_motor), [generators](/source/Electric_generator), [transformers](/source/Transformer), [relays](/source/Relay), lifting [electromagnets](/source/Electromagnet), [SQUIDs](/source/SQUID), [galvanometers](/source/Galvanometer), and magnetic [recording heads](/source/Recording_head).

The relation between the magnetic properties of a magnetic circuit can be described by [Hopkinson's law](/source/Hopkinson's_law), which bears a superficial resemblance to [Ohm's law](/source/Ohm's_law) in electrical circuits, resulting in a one-to-one correspondence between properties of a magnetic circuit and an analogous electric circuit. Using this concept the magnetic fields of complex devices such as transformers can be quickly solved using the methods and techniques developed for electrical circuits. Hopkinson's law is:[82]

**Hopkinson's Law**

Φ = F R m , {\displaystyle \Phi ={\frac {F}{R}}_{\mathrm {m} },}

where Φ = ∫ B ⋅ d A {\textstyle \Phi =\int \mathbf {B} \cdot \mathrm {d} \mathbf {A} } is the [magnetic flux](/source/Magnetic_flux) in the circuit, F = ∫ H ⋅ d ℓ {\textstyle F=\int \mathbf {H} \cdot \mathrm {d} {\boldsymbol {\ell }}} is the [magnetomotive force](/source/Magnetomotive_force) applied to the circuit, and *R*m is the [magnetic reluctance](/source/Magnetic_reluctance) of the circuit. Here the reluctance *R*m is a quantity similar in nature to [resistance](/source/Electrical_resistance) for the flux. Using this analogy it is straightforward to calculate the magnetic flux of complicated magnetic field geometries, by using all the available techniques of [circuit theory](/source/Circuit_theory).

#### Magnetic levitation

The [Transrapid](/source/Transrapid) system uses servomechanisms to pull the train up from underneath the track and maintains a constant gap while travelling at high speed

Main article: [magnetic levitation](/source/Magnetic_levitation)

**Magnetic levitation** (**maglev**) or **magnetic suspension** is a method by which an object is [suspended](/source/Levitation_(physics)) with no support other than magnetic fields. [Magnetic force](/source/Lorentz_force) is used to counteract the effects of the [gravitational force](/source/Gravitational_force) and any other forces.[83] The two primary issues involved in magnetic levitation are (a) lifting forces – providing an upward force sufficient to counteract gravity, and (b) stability – ensuring that the system does not spontaneously slide or flip into a configuration where the lift is neutralized.

Magnetic levitation is used for [maglev](/source/Maglev) trains, [contactless melting](/source/Levitation_melting), [magnetic bearings](/source/Magnetic_bearing), and for product display purposes.

### Uses in material science

Main article: [List of materials properties § Magnetic properties](/source/List_of_materials_properties#Magnetic_properties)

Magnetic field affects materials in a large number of ways.

#### Hall effect

Main article: [Hall effect](/source/Hall_effect)

The charge carriers of a current-carrying conductor placed in a transverse magnetic field experience a sideways Lorentz force; this results in a charge separation in a direction perpendicular to the current and to the magnetic field. The resultant voltage in that direction is proportional to the applied magnetic field. This is known as the *Hall effect*.

The *Hall effect* is often used to measure the magnitude of a magnetic field. It is used as well to find the sign of the dominant charge carriers in materials such as semiconductors (negative electrons or positive holes).

### Largest magnitude magnetic fields

Main article: [Orders of magnitude (magnetic field)](/source/Orders_of_magnitude_(magnetic_field))

The largest magnitude magnetic field produced over a macroscopic volume outside a lab setting is 2.8 kT ([VNIIEF](/source/VNIIEF) in [Sarov](/source/Sarov), [Russia](/source/Russia), 1998).[84] The largest magnitude magnetic field produced in a laboratory over a macroscopic volume was 1.2 kT by researchers at the [University of Tokyo](/source/University_of_Tokyo) in 2018.[85] The largest magnitude microscopic magnetic fields produced in a laboratory occur in particle accelerators, such as [RHIC](/source/Relativistic_Heavy_Ion_Collider), inside the collisions of heavy ions, where microscopic fields reach 1014 T.[86][87] [Magnetars](/source/Magnetar) have the strongest known macroscopic magnetic fields of any naturally occurring object, ranging from 0.1 to 100 GT (108 to 1011 T).[88]

## Common formulae

Current configuration Figure Magnetic field Finite beam of current B = μ 0 I 4 π x ( cos ⁡ θ 1 + cos ⁡ θ 2 ) {\displaystyle B={\frac {\mu _{0}I}{4\pi x}}(\cos \theta _{1}+\cos \theta _{2})} where I {\displaystyle I} is the uniform current throughout the beam, with the direction of magnetic field as shown. Infinite wire B = μ 0 I 2 π x {\displaystyle B={\frac {\mu _{0}I}{2\pi x}}} where I {\displaystyle I} is the uniform current flowing through the wire with the direction of magnetic field as shown. Infinite cylindrical wire B = μ 0 I 2 π x {\displaystyle B={\frac {\mu _{0}I}{2\pi x}}} outside the wire carrying a current I {\displaystyle I} uniformly, with the direction of magnetic field as shown. B = μ 0 I x 2 π R 2 {\displaystyle B={\frac {\mu _{0}Ix}{2\pi R^{2}}}} inside the wire carrying a current I {\displaystyle I} uniformly, with the direction of magnetic field as shown. Circular loop B = μ 0 I R 2 2 ( x 2 + R 2 ) 3 / 2 x ^ {\displaystyle \mathbf {B} ={\frac {\mu _{0}IR^{2}}{2(x^{2}+R^{2})^{3/2}}}{\hat {\mathbf {x} }}} along the axis of the loop, where I {\displaystyle I} is the uniform current flowing through the loop. Solenoid B = μ 0 n I 2 ( cos ⁡ θ 1 + cos ⁡ θ 2 ) {\displaystyle B={\frac {\mu _{0}nI}{2}}(\cos \theta _{1}+\cos \theta _{2})} along the axis of the solenoid carrying current I {\displaystyle I} with n {\displaystyle n} , uniform number of loops of currents per length of solenoid; and the direction of magnetic field as shown. Infinite solenoid B = 0 {\displaystyle \mathbf {B} =0} outside the solenoid carrying current I {\displaystyle I} with n {\displaystyle n} , uniform number of loops of currents per length of solenoid. B = μ 0 n I {\displaystyle B=\mu _{0}nI} inside the solenoid carrying current I {\displaystyle I} with n {\displaystyle n} , uniform number of loops of currents per length of solenoid, with the direction of magnetic field as shown. Circular Toroid B = μ 0 N I 2 π R {\displaystyle B={\frac {\mu _{0}NI}{2\pi R}}} along the bulk of the circular toroid carrying uniform current I {\displaystyle I} through N {\displaystyle N} number of uniformly distributed poloidal loops, with the direction of magnetic field as indicated. Magnetic Dipole B = − μ 0 m 4 π r 3 , {\displaystyle \mathbf {B} =-{\frac {\mu _{0}\mathbf {m} }{4\pi r^{3}}},} on the equatorial plane, where m {\displaystyle \mathbf {m} } is the magnetic dipole moment. B = μ 0 m 2 π | x | 3 , {\displaystyle \mathbf {B} ={\frac {\mu _{0}\mathbf {m} }{2\pi {|x|}^{3}}},} on the axial plane (given that x ≫ R {\displaystyle x\gg R} ), where x {\displaystyle x} can also be negative to indicate position at the opposite direction on the axis, and m {\displaystyle \mathbf {m} } is the magnetic dipole moment.

Additional magnetic field values can be found through the magnetic field of a finite beam, for example, that the magnetic field of an arc of angle θ {\displaystyle \theta } and radius R {\displaystyle R} at the center is B = μ 0 θ I 4 π R {\displaystyle B={\mu _{0}\theta I \over 4\pi R}} , or that the magnetic field at the center of a N-sided regular polygon of side a {\displaystyle a} is B = μ 0 N I π a sin ⁡ π N tan ⁡ π N {\displaystyle B={\mu _{0}NI \over \pi a}\sin {\pi \over N}\tan {\pi \over N}} , both outside of the plane with proper directions as inferred by right hand thumb rule.

## History

Main article: [History of electromagnetic theory](/source/History_of_electromagnetic_theory)

See also: [Timeline of electromagnetism and classical optics](/source/Timeline_of_electromagnetism_and_classical_optics)

One of the first drawings of a magnetic field, by [René Descartes](/source/Ren%C3%A9_Descartes), 1644, showing the Earth attracting [lodestones](/source/Lodestone). It illustrated his theory that magnetism was caused by the circulation of tiny helical particles, "threaded parts", through threaded pores in magnets.

### Early developments

While magnets and some properties of magnetism were known to ancient societies, the research of magnetic fields began in 1269 when French scholar [Petrus Peregrinus de Maricourt](/source/Petrus_Peregrinus_de_Maricourt) mapped out the magnetic field on the surface of a spherical magnet using iron needles. Noting the resulting field lines crossed at two points he named those points "poles" in analogy to Earth's poles. He also articulated the principle that magnets always have both a north and south pole, no matter how finely one slices them.[89][note 14]

In 1600 (almost three centuries later), [William Gilbert](/source/William_Gilbert_(astronomer)) of [Colchester](/source/Colchester) published *[De Magnete](/source/De_Magnete)*. In *De Magnete*, Gilbert replicated Petrus Peregrinus' work and was the first to state explicitly that Earth is a magnet.[90]: 34 Too, he argued that electricity and magnetism were separate phenomenon.

### Magnetostatics

[Hans Christian Ørsted](/source/Hans_Christian_%C3%98rsted), *Der Geist in der Natur*, 1854

In 1750, [John Michell](/source/John_Michell) stated that magnetic poles attract and repel in accordance with an [inverse square law](/source/Inverse_square_law)[90]: 56 [Charles-Augustin de Coulomb](/source/Charles-Augustin_de_Coulomb) experimentally verified this in 1785 and stated explicitly that north and south poles cannot be separated.[90]: 59 Building on this force between poles, [Siméon Denis Poisson](/source/Sim%C3%A9on_Denis_Poisson) (1781–1840) created the first successful model of the magnetic field, which he presented in 1824.[90]: 64

Three discoveries in 1820 challenged this foundation of magnetism. [Hans Christian Ørsted](/source/Hans_Christian_%C3%98rsted) demonstrated that a current-carrying wire is surrounded by a circular magnetic field.[note 15][91] Then [André-Marie Ampère](/source/Andr%C3%A9-Marie_Amp%C3%A8re) showed that parallel wires with currents attract one another if the currents are in the same direction and repel if they are in opposite directions.[90]: 87[92] Finally, [Jean-Baptiste Biot](/source/Jean-Baptiste_Biot) and [Félix Savart](/source/F%C3%A9lix_Savart) announced empirical results about the forces that a current-carrying long, straight wire exerted on a small magnet, determining the forces were inversely proportional to the perpendicular distance from the wire to the magnet.[93][90]: 86 [Laplace](/source/Laplace) later deduced a law of force based on the differential action of a differential section of the wire,[93][94] which became known as the [Biot–Savart law](/source/Biot%E2%80%93Savart_law), as Laplace did not publish his findings.[95]

Extending these experiments, Ampère published his own successful model of magnetism in 1825. In it, he showed the equivalence of electrical currents to magnets[90]: 88 and proposed that magnetism is due to perpetually flowing loops of current instead of the dipoles of magnetic charge in Poisson's model.[note 16] Further, Ampère derived both [Ampère's force law](/source/Amp%C3%A8re's_force_law) describing the force between two currents and [Ampère's law](/source/Amp%C3%A8re's_law), which, like the Biot–Savart law, correctly described the magnetic field generated by a steady current.

### Electrodynamics

Main article: [History of Maxwell's equations](/source/History_of_Maxwell's_equations)

Also in his 1825 work, Ampère introduced the term [electrodynamics](/source/Electrodynamics) to describe the relationship between electricity and magnetism.[90]: 88–92

In 1831, [Michael Faraday](/source/Michael_Faraday) discovered [electromagnetic induction](/source/Electromagnetic_induction) when he found that a changing magnetic field generates an encircling electric field, formulating what is now known as [Faraday's law of induction](/source/Faraday's_law_of_induction).[90]: 189–192 Later, [Franz Ernst Neumann](/source/Franz_Ernst_Neumann) proved that, for a moving conductor in a magnetic field, induction is a consequence of Ampère's force law.[90]: 222 In the process, he introduced the magnetic vector potential, which was later shown to be equivalent to the underlying mechanism proposed by Faraday.[90]: 225 In 1850, [Lord Kelvin](/source/Lord_Kelvin), then known as William Thomson, distinguished between two magnetic fields now denoted **H** and **B**. The former applied to Poisson's model and the latter to Ampère's model and induction.[90]: 224 Further, he derived how **H** and **B** relate to each other and coined the term *permeability*.[90]: 245[96]

Between 1861 and 1865, [James Clerk Maxwell](/source/James_Clerk_Maxwell) developed and published [Maxwell's equations](/source/Maxwell's_equations), which explained and united all of [classical](/source/Classical_theory) electricity and magnetism. The first set of these equations was published in a paper entitled *[On Physical Lines of Force](/source/On_Physical_Lines_of_Force)* in 1861. These equations were valid but incomplete. Maxwell completed his set of equations in his later 1865 paper *[A Dynamical Theory of the Electromagnetic Field](/source/A_Dynamical_Theory_of_the_Electromagnetic_Field)* and demonstrated the fact that light is an [electromagnetic wave](/source/Electromagnetic_wave). [Heinrich Hertz](/source/Heinrich_Hertz) published papers in 1887 and 1888 experimentally confirming this fact.[97][98]

### Modern developments

Main articles: [History of electrical engineering](/source/History_of_electrical_engineering) and [History of quantum field theory](/source/History_of_quantum_field_theory)

In 1887, Tesla developed an [induction motor](/source/Induction_motor) that ran on [alternating current](/source/Alternating_current). The motor used [polyphase](/source/Polyphase_system) current, which generated a [rotating magnetic field](/source/Rotating_magnetic_field) to turn the motor (a principle that Tesla claimed to have conceived in 1882).[99][100][101] Tesla received a patent for his electric motor in May 1888.[102][103] In 1885, [Galileo Ferraris](/source/Galileo_Ferraris) independently researched rotating magnetic fields and subsequently published his research in a paper to the *Royal Academy of Sciences* in [Turin](/source/Turin), just two months before Tesla was awarded his patent, in March 1888.[104]

The twentieth century showed that classical electrodynamics is already consistent with special relativity, and extended classical electrodynamics to work with quantum mechanics. [Albert Einstein](/source/Albert_Einstein), in his paper of 1905 that established relativity, showed that both the electric and magnetic fields are part of the same phenomena viewed from different reference frames. Finally, the emergent field of [quantum mechanics](/source/Quantum_mechanics) was merged with electrodynamics to form [quantum electrodynamics](/source/Quantum_electrodynamics) (or QED). QED mathematically describes all [phenomena](/source/Phenomenon) involving [electrically charged](/source/Electric_charge) particles interacting by means of exchange of [photons](/source/Photon) and represents the [quantum](/source/Quantum_mechanics) counterpart of [classical electromagnetism](/source/Classical_electromagnetism) giving a complete account of matter and light interaction.[105]

## Links, references, and notes

### See also

#### General

- [Magnetohydrodynamics](/source/Magnetohydrodynamics) – the study of the dynamics of electrically conducting fluids

- [Magnetic hysteresis](/source/Magnetic_hysteresis) – application to [ferromagnetism](/source/Ferromagnetism)

- [Magnetic nanoparticles](/source/Magnetic_nanoparticles) – extremely small magnetic particles that are tens of atoms wide

- [Magnetic reconnection](/source/Magnetic_reconnection) – an effect that causes [solar flares](/source/Solar_flare) and auroras

- [Magnetic scalar potential](/source/Magnetic_scalar_potential)

- [SI electromagnetism units](/source/SI_electromagnetism_units) – common units used in electromagnetism

- [Orders of magnitude (magnetic field)](/source/Orders_of_magnitude_(magnetic_field)) – list of magnetic field sources and measurement devices from smallest magnetic fields to largest detected

- [Upward continuation](/source/Upward_continuation)

- [Moses Effect](/source/Moses_Effect)

#### Mathematics

- [Magnetic helicity](/source/Magnetic_helicity) – extent to which a magnetic field wraps around itself

#### Applications

- [Dynamo theory](/source/Dynamo_theory) – a proposed mechanism for the creation of the Earth's magnetic field

- [Helmholtz coil](/source/Helmholtz_coil) – a device for producing a region of nearly uniform magnetic field

- [Magnetic field viewing film](/source/Magnetic_field_viewing_film) – Film used to view the magnetic field of an area

- [Magnetic pistol](/source/Magnetic_pistol) – a device on torpedoes or naval mines that detect the magnetic field of their target

- [Maxwell coil](/source/Maxwell_coil) – a device for producing a large volume of an almost constant magnetic field

- [Stellar magnetic field](/source/Stellar_magnetic_field) – a discussion of the magnetic field of stars

- [Teltron tube](/source/Teltron_tube) – device used to display an electron beam and demonstrates effect of electric and magnetic fields on moving charges

### Notes

1. **[^](#cite_ref-1)** More precisely, magnetic field is a [pseudovector](/source/Pseudovector) field due to its properties under inversion.

1. **[^](#cite_ref-2)** The letters B and H were originally chosen by Maxwell in his *[Treatise on Electricity and Magnetism](/source/Treatise_on_Electricity_and_Magnetism)* (Vol. II, pp. 236–237). For many quantities, he simply started choosing letters from the beginning of the alphabet. See Ralph Baierlein (2000). "Answer to Question #73. S is for entropy, Q is for charge". *American Journal of Physics*. **68** (8): 691. [Bibcode](/source/Bibcode_(identifier)):[2000AmJPh..68..691B](https://ui.adsabs.harvard.edu/abs/2000AmJPh..68..691B). [doi](/source/Doi_(identifier)):[10.1119/1.19524](https://doi.org/10.1119%2F1.19524).

1. **[^](#cite_ref-11)** The SI unit of Φ*B* ([magnetic flux](/source/Magnetic_flux)) is the [weber](/source/Weber_(unit)) (symbol: Wb), related to the [tesla](/source/Tesla_(unit)) by 1 Wb/m2 = 1 T. The SI unit tesla is equal to ([newton](/source/Newton_(unit))·[second](/source/Second))/([coulomb](/source/Coulomb_(unit))·[metre](/source/Metre)). This can be seen from the magnetic part of the Lorentz force law.

1. **[^](#cite_ref-16)** [Griffiths 1999](#CITEREFGriffiths1999), p. 272 "As it turns out, **H** is a more useful quantity than **D**. ... The reason is this: To build an electromagnet you run a certain (free) current through a coil. The *current* is the thing you read on the dial, and this determines **H** (or at any rate, the line integral of **H**)."

1. **[^](#cite_ref-19)** The induced component is zero for certain high-symmetry cases where Ampere's law can easily be used and is irrelevant for certain quantities such as those that depend a line integral (over a loop) of the **H**-field such as in the [MMF](/source/Magnetomotive_force) of [magnetic circuits](/source/Magnetic_circuits).

1. **[^](#cite_ref-20)** This component of H is called the [demagnetizing field](/source/Demagnetizing_field), or stray field

1. **[^](#cite_ref-ex12_37-0)** In practice, the Biot–Savart law and other laws of magnetostatics are often used even when a current change in time, as long as it does not change too quickly. It is often used, for instance, for standard household currents, which oscillate sixty times per second.[30]

1. **[^](#cite_ref-ex13_46-0)** The Biot–Savart law contains the additional restriction (boundary condition) that the B-field must go to zero fast enough at infinity. It also depends on the divergence of **B** being zero, which is always valid. (There are no magnetic charges.)

1. **[^](#cite_ref-49)** The **H**-field calculated this way does not include that due to magnetic poles which is called the stray field or [demagnetizing field](/source/Demagnetizing_field) and must be calculated separately if needed.

1. **[^](#cite_ref-ex04_50-0)** Either **B** or **H** may be used for the magnetic field outside the magnet.

1. **[^](#cite_ref-61)** [Ferrimagnetic](/source/Ferrimagnetism) materials, such as [magnetite](/source/Magnetite), can also be magnetized.

1. **[^](#cite_ref-68)** A third term is needed for changing electric fields and polarization currents; this displacement current term is covered in Maxwell's equations below.

1. **[^](#cite_ref-79)** See [Helmholtz decomposition#Three-dimensional space](/source/Helmholtz_decomposition#Three-dimensional_space).

1. **[^](#cite_ref-103)** His *Epistola Petri Peregrini de Maricourt ad Sygerum de Foucaucourt Militem de Magnete*, which is often shortened to *Epistola de magnete*, is dated 1269 C.E.

1. **[^](#cite_ref-105)** During a lecture demonstration on the effects of a current on a campus needle, Ørsted showed that when a current-carrying wire is placed at a right angle with the compass, nothing happens. When he tried to orient the wire parallel to the compass needle, however, it produced a pronounced deflection of the compass needle. By placing the compass on different sides of the wire, he was able to determine the field forms perfect circles around the wire.[90]: 85

1. **[^](#cite_ref-111)** From the outside, the field of a dipole of magnetic charge has exactly the same form as a current loop when both are sufficiently small. Therefore, the two models differ only for magnetism inside magnetic material.

### References

1. **[^](#cite_ref-3)** John J. Roche (2000). "B and H, the intensity vectors of magnetism: A new approach to resolving a century-old controversy". *American Journal of Physics*. **68** (5): 438. [Bibcode](/source/Bibcode_(identifier)):[2000AmJPh..68..438R](https://ui.adsabs.harvard.edu/abs/2000AmJPh..68..438R). [doi](/source/Doi_(identifier)):[10.1119/1.19459](https://doi.org/10.1119%2F1.19459).

1. **[^](#cite_ref-4)** ["ISO 80000-6:2022(en,fr) Quantities and units – Part 6: Electromagnetism"](https://www.iso.org/obp/ui/#iso:std:iec:80000:-6:ed-2:v1:en,fr). [International Organization for Standardization](/source/International_Organization_for_Standardization). Retrieved 21 November 2022. [\[1\]](https://www.electropedia.org/iev/iev.nsf/display?openform&ievref=221-01-01)

1. **[^](#cite_ref-5)** [*The International System of Units*](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf) (PDF), V4.01 (9th ed.), International Bureau of Weights and Measures, June 2026, p. 138, [ISBN](/source/ISBN_(identifier)) [978-92-822-2272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-92-822-2272-0)

1. ^ [***a***](#cite_ref-FOOTNOTEStratton19411_6-0) [***b***](#cite_ref-FOOTNOTEStratton19411_6-1) [Stratton 1941](#CITEREFStratton1941), p. 1.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999203_7-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 203.

1. ^ [***a***](#cite_ref-FOOTNOTEPurcell2011173_8-0) [***b***](#cite_ref-FOOTNOTEPurcell2011173_8-1) [Purcell 2011](#CITEREFPurcell2011), p. 173.

1. ^ [***a***](#cite_ref-FOOTNOTEGriffiths1999204Equation_5.1_9-0) [***b***](#cite_ref-FOOTNOTEGriffiths1999204Equation_5.1_9-1) [Griffiths 1999](#CITEREFGriffiths1999), p. 204, Equation 5.1.

1. **[^](#cite_ref-FOOTNOTEPurcell2011pp._[httpsarchiveorgdetailselectricitymagne00purc_621pagen192_173]–4_10-0)** [Purcell 2011](#CITEREFPurcell2011), pp. [173](https://archive.org/details/electricitymagne00purc_621/page/n192)–4.

1. **[^](#cite_ref-12)** [Purcell 2011](#CITEREFPurcell2011), p. 286 Tesla for describing a large magnetic force; gauss (tesla/10000) for describing a small magnetic force as that at the surface of earth.

1. ^ [***a***](#cite_ref-BIPMTab9_13-0) [***b***](#cite_ref-BIPMTab9_13-1) ["Non-SI units accepted for use with the SI, and units based on fundamental constants (contd.)"](https://web.archive.org/web/20190608123210/https://www.bipm.org/en/publications/si-brochure/table9.html). *SI Brochure: The International System of Units (SI) [8th edition, 2006; updated in 2014]*. Bureau International des Poids et Mesures. Archived from [the original](https://www.bipm.org/en/publications/si-brochure/table9.html) on 8 June 2019. Retrieved 19 April 2018.

1. ^ [***a***](#cite_ref-KLang_14-0) [***b***](#cite_ref-KLang_14-1) Lang, Kenneth R. (2006). [*A Companion to Astronomy and Astrophysics*](https://books.google.com/books?id=aUjkKuaVIloC&pg=PA176). Springer. p. 176. [ISBN](/source/ISBN_(identifier)) [978-0-387-33367-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-33367-0). Retrieved 19 April 2018.

1. **[^](#cite_ref-15)** [*The International System of Units*](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf) (PDF), V4.01 (9th ed.), International Bureau of Weights and Measures, June 2026, p. 138, [ISBN](/source/ISBN_(identifier)) [978-92-822-2272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-92-822-2272-0)

1. **[^](#cite_ref-17)** [*The International System of Units*](https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf) (PDF), V4.01 (9th ed.), International Bureau of Weights and Measures, June 2026, p. 139, [ISBN](/source/ISBN_(identifier)) [978-92-822-2272-0](https://en.wikipedia.org/wiki/Special:BookSources/978-92-822-2272-0)

1. **[^](#cite_ref-FOOTNOTEGriffiths1999269Equation_6.18_18-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 269, Equation 6.18.

1. **[^](#cite_ref-21)** ["International system of units (SI)"](https://physics.nist.gov/cuu/Units/units.html). *NIST reference on constants, units, and uncertainty*. National Institute of Standards and Technology. 12 April 2010. Retrieved 9 May 2012.

1. **[^](#cite_ref-22)** ["Gravity Probe B Executive Summary"](https://www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf) (PDF). pp. 10, 21. [Archived](https://ghostarchive.org/archive/20221009/http://www.nasa.gov/pdf/168808main_gp-b_pfar_cvr-pref-execsum.pdf) (PDF) from the original on 9 October 2022.

1. **[^](#cite_ref-FOOTNOTELingMoebsSanny2016§_11.2_23-0)** [Ling, Moebs & Sanny 2016](#CITEREFLingMoebsSanny2016), § 11.2.

1. **[^](#cite_ref-FOOTNOTEPurcell2011237_24-0)** [Purcell 2011](#CITEREFPurcell2011), p. 237.

1. **[^](#cite_ref-FOOTNOTEPurcellMorin2013284_25-0)** [Purcell & Morin 2013](#CITEREFPurcellMorin2013), p. 284.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999209Eq._5.16_26-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 209, Eq. 5.16.

1. **[^](#cite_ref-FOOTNOTELingMoebsSanny2016§_11.5_27-0)** [Ling, Moebs & Sanny 2016](#CITEREFLingMoebsSanny2016), § 11.5.

1. **[^](#cite_ref-FOOTNOTELingMoebsSanny2016Eq._11.19_28-0)** [Ling, Moebs & Sanny 2016](#CITEREFLingMoebsSanny2016), Eq. 11.19.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999257_29-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 257.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999244Eq._5.84_30-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 244, Eq. 5.84.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999257Eq._6.1_31-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 257, Eq. 6.1.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999258Eq._6.3_32-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 258, Eq. 6.3.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999438_33-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 438.

1. **[^](#cite_ref-FOOTNOTEIda2021383Example_8.1c_34-0)** [Ida 2021](#CITEREFIda2021), p. 383, Example 8.1c.

1. **[^](#cite_ref-FOOTNOTEGriffiths2017225_35-0)** [Griffiths 2017](#CITEREFGriffiths2017), p. 225.

1. **[^](#cite_ref-FOOTNOTEGriffiths2017223_36-0)** [Griffiths 2017](#CITEREFGriffiths2017), p. 223.

1. **[^](#cite_ref-FOOTNOTEGriffiths2017224_38-0)** [Griffiths 2017](#CITEREFGriffiths2017), p. 224.

1. **[^](#cite_ref-FOOTNOTEIda2021381Eq._8.8_39-0)** [Ida 2021](#CITEREFIda2021), p. 381, Eq. 8.8.

1. **[^](#cite_ref-FOOTNOTEIda2021394Example_8.10_40-0)** [Ida 2021](#CITEREFIda2021), p. 394, Example 8.10.

1. **[^](#cite_ref-FOOTNOTELingMoebsSanny2016Eq._12.16_in_§_12.4_41-0)** [Ling, Moebs & Sanny 2016](#CITEREFLingMoebsSanny2016), Eq. 12.16 in § 12.4.

1. **[^](#cite_ref-FOOTNOTEIda2021384–5Example_8.3_42-0)** [Ida 2021](#CITEREFIda2021), pp. 384–5, Example 8.3.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999242–6§_5.4.3_43-0)** [Griffiths 1999](#CITEREFGriffiths1999), pp. 242–6, § 5.4.3.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999246Eq._5.87_44-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 246, Eq. 5.87.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999222–225_45-0)** [Griffiths 1999](#CITEREFGriffiths1999), pp. 222–225.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999225Eq._5.55_47-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 225, Eq. 5.55.

1. **[^](#cite_ref-FOOTNOTEIda2021389Eq._8.16_48-0)** [Ida 2021](#CITEREFIda2021), p. 389, Eq. 8.16.

1. **[^](#cite_ref-51)** ["Basic Relationships"](https://web.archive.org/web/20100709205321/http://geophysics.ou.edu/solid_earth/notes/mag_basic/mag_basic.html). Geophysics.ou.edu. Archived from [the original](http://geophysics.ou.edu/solid_earth/notes/mag_basic/mag_basic.html) on 9 July 2010. Retrieved 19 October 2009.

1. **[^](#cite_ref-tri-c_52-0)** ["Magnetic Fields and Forces"](https://web.archive.org/web/20120220030524/http://instruct.tri-c.edu/fgram/web/Mdipole.htm). Archived from [the original](http://instruct.tri-c.edu/fgram/web/Mdipole.htm) on 20 February 2012. Retrieved 24 December 2009.

1. **[^](#cite_ref-53)** ["The force produced by a magnetic field"](https://web.archive.org/web/20100317153105/http://info.ee.surrey.ac.uk/Workshop/advice/coils/force.html). Archived from [the original](http://info.ee.surrey.ac.uk/Workshop/advice/coils/force.html) on 17 March 2010. Retrieved 7 November 2013.

1. **[^](#cite_ref-54)** ["Tutorial: Theory and applications of the Maxwell stress tenso"](https://www.fieldp.com/documents/stresstensor.pdf) (PDF). Retrieved 28 November 2018.

1. **[^](#cite_ref-FOOTNOTEIda2021483Eq._9.127_55-0)** [Ida 2021](#CITEREFIda2021), p. 483, Eq. 9.127.

1. ^ [***a***](#cite_ref-Tilley_56-0) [***b***](#cite_ref-Tilley_56-1) RJD Tilley (2004). [*Understanding Solids*](https://archive.org/details/understandingsol0000till). Wiley. p. [368](https://archive.org/details/understandingsol0000till/page/368). [ISBN](/source/ISBN_(identifier)) [978-0-470-85275-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-470-85275-0).

1. **[^](#cite_ref-Chikazumi_57-0)** Sōshin Chikazumi; Chad D. Graham (1997). [*Physics of ferromagnetism*](https://books.google.com/books?id=AZVfuxXF2GsC) (2 ed.). Oxford University Press. p. 118. [ISBN](/source/ISBN_(identifier)) [978-0-19-851776-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-851776-4).

1. **[^](#cite_ref-Aharoni_58-0)** Amikam Aharoni (2000). [*Introduction to the theory of ferromagnetism*](https://books.google.com/books?id=9RvNuIDh0qMC&pg=PA27) (2 ed.). Oxford University Press. p. 27. [ISBN](/source/ISBN_(identifier)) [978-0-19-850808-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-850808-3).

1. **[^](#cite_ref-Bennemann_59-0)** M Brian Maple; et al. (2008). ["Unconventional superconductivity in novel materials"](https://books.google.com/books?id=PguAgEQTiQwC&pg=PA640). In K. H. Bennemann; John B. Ketterson (eds.). *Superconductivity*. Springer. p. 640. [ISBN](/source/ISBN_(identifier)) [978-3-540-73252-5](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-73252-5).

1. **[^](#cite_ref-Lewis_60-0)** Naoum Karchev (2003). ["Itinerant ferromagnetism and superconductivity"](https://books.google.com/books?id=3AFo_yxBkD0C&pg=PA169). In Paul S. Lewis; D. Di (CON) Castro (eds.). *Superconductivity research at the leading edge*. Nova Publishers. p. 169. [ISBN](/source/ISBN_(identifier)) [978-1-59033-861-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-59033-861-2).

1. **[^](#cite_ref-FOOTNOTEIda2021426§9.2.2_Eq._9.17_62-0)** [Ida 2021](#CITEREFIda2021), p. 426, §9.2.2 Eq. 9.17.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999266–268_63-0)** [Griffiths 1999](#CITEREFGriffiths1999), pp. 266–268.

1. **[^](#cite_ref-64)** This equation is analogous to the integral form of [Griffiths 1999](#CITEREFGriffiths1999), p. 168

1. **[^](#cite_ref-FOOTNOTEIda2021433Eq._9.30_65-0)** [Ida 2021](#CITEREFIda2021), p. 433, Eq. 9.30.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999269§6.3.1,_Eq._6.20_66-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 269, §6.3.1, Eq. 6.20.

1. **[^](#cite_ref-Slater_67-0)** John Clarke Slater; Nathaniel Herman Frank (1969). [*Electromagnetism*](https://books.google.com/books?id=GYsphnFwUuUC&pg=PA69) (first published in 1947 ed.). Courier Dover Publications. p. 69. [ISBN](/source/ISBN_(identifier)) [978-0-486-62263-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-62263-7).

1. **[^](#cite_ref-FOOTNOTEIda2021380Eq._8.3_69-0)** [Ida 2021](#CITEREFIda2021), p. 380, Eq. 8.3.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999332_70-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 332.

1. **[^](#cite_ref-FOOTNOTEIda442_71-0)** [Ida](#CITEREFIda), p. 442. sfn error: no target: CITEREFIda ([help](https://en.wikipedia.org/wiki/Category:Harv_and_Sfn_template_errors))

1. **[^](#cite_ref-FOOTNOTEGriffiths1999296Eq._7.13_72-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 296, Eq. 7.13.

1. **[^](#cite_ref-FOOTNOTEJackson1975210_73-0)** [Jackson 1975](#CITEREFJackson1975), p. 210.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999318Eq._7.34_74-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 318, Eq. 7.34.

1. **[^](#cite_ref-FOOTNOTEIda2021604Eq,_12.52_75-0)** [Ida 2021](#CITEREFIda2021), p. 604, Eq, 12.52.

1. **[^](#cite_ref-FOOTNOTEWangsness1986334Eq._20-79_76-0)** [Wangsness 1986](#CITEREFWangsness1986), p. 334, Eq. 20-79.

1. **[^](#cite_ref-77)** [Poynting, John Henry](/source/John_Henry_Poynting) (1884). ["On the Transfer of Energy in the Electromagnetic Field"](https://doi.org/10.1098%2Frstl.1884.0016). *Philosophical Transactions of the Royal Society of London*. **175** (175): 343–361. [Bibcode](/source/Bibcode_(identifier)):[1884RSPT..175..343.](https://ui.adsabs.harvard.edu/abs/1884RSPT..175..343.). [doi](/source/Doi_(identifier)):[10.1098/rstl.1884.0016](https://doi.org/10.1098%2Frstl.1884.0016).

1. **[^](#cite_ref-FOOTNOTEIda2021604Eq._12.53_78-0)** [Ida 2021](#CITEREFIda2021), p. 604, Eq. 12.53.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999416–7Eqs._10.2_&_10.3_80-0)** [Griffiths 1999](#CITEREFGriffiths1999), pp. 416–7, Eqs. 10.2 & 10.3.

1. **[^](#cite_ref-81)** [E. J. Konopinski](/source/Emil_Konopinski) (1978). "What the electromagnetic vector potential describes". *Am. J. Phys*. **46** (5): 499–502. [Bibcode](/source/Bibcode_(identifier)):[1978AmJPh..46..499K](https://ui.adsabs.harvard.edu/abs/1978AmJPh..46..499K). [doi](/source/Doi_(identifier)):[10.1119/1.11298](https://doi.org/10.1119%2F1.11298).

1. **[^](#cite_ref-FOOTNOTEGriffiths1999Eqs._10.15_&_10.16_82-0)** [Griffiths 1999](#CITEREFGriffiths1999), Eqs. 10.15 & 10.16.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999422_83-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 422.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999&sect._12.3.2_84-0)** [Griffiths 1999](#CITEREFGriffiths1999), &sect. 12.3.2.

1. **[^](#cite_ref-Rosser1968_85-0)** Rosser, W. G. V. (1968). [*Classical Electromagnetism via Relativity*](https://link.springer.com/book/10.1007/978-1-4899-6559-2). Boston, MA: Springer. [doi](/source/Doi_(identifier)):[10.1007/978-1-4899-6559-2](https://doi.org/10.1007%2F978-1-4899-6559-2). [ISBN](/source/ISBN_(identifier)) [978-1-4899-6258-4](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4899-6258-4).

1. **[^](#cite_ref-86)** C. Doran and A. Lasenby (2003) *Geometric Algebra for Physicists*, Cambridge University Press, p. 233. [ISBN](/source/ISBN_(identifier)) [0-521-71595-4](https://en.wikipedia.org/wiki/Special:BookSources/0-521-71595-4).

1. **[^](#cite_ref-FOOTNOTEGriffiths1999435Eq._10.47_87-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 435, Eq. 10.47.

1. **[^](#cite_ref-FOOTNOTEGriffiths1999438Eqs._10.64,_10.65,_&_10.66_88-0)** [Griffiths 1999](#CITEREFGriffiths1999), p. 438, Eqs. 10.64, 10.65, & 10.66.

1. **[^](#cite_ref-89)** For a good qualitative introduction see: [Richard Feynman](/source/Richard_Feynman) (2006). [*QED: the strange theory of light and matter*](/source/QED_(book)). [Princeton University Press](/source/Princeton_University_Press). [ISBN](/source/ISBN_(identifier)) [978-0-691-12575-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-691-12575-6).

1. **[^](#cite_ref-Weiss_90-0)** Weiss, Nigel (2002). ["Dynamos in planets, stars and galaxies"](https://doi.org/10.1046%2Fj.1468-4004.2002.43309.x). *Astronomy and Geophysics*. **43** (3): 3.09 – 3.15. [Bibcode](/source/Bibcode_(identifier)):[2002A&G....43c...9W](https://ui.adsabs.harvard.edu/abs/2002A&G....43c...9W). [doi](/source/Doi_(identifier)):[10.1046/j.1468-4004.2002.43309.x](https://doi.org/10.1046%2Fj.1468-4004.2002.43309.x).

1. **[^](#cite_ref-91)** ["What is the Earth's magnetic field?"](https://www.ncei.noaa.gov/products/geomagnetism-frequently-asked-questions). *Geomagnetism Frequently Asked Questions*. National Centers for Environmental Information, National Oceanic and Atmospheric Administration. Retrieved 19 April 2018.

1. **[^](#cite_ref-92)** Raymond A. Serway; Chris Vuille; Jerry S. Faughn (2009). [*College physics*](https://archive.org/details/collegephysics00serw_139) (8th ed.). Belmont, CA: Brooks/Cole, Cengage Learning. p. [628](https://archive.org/details/collegephysics00serw_139/page/n659). [ISBN](/source/ISBN_(identifier)) [978-0-495-38693-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-495-38693-3).

1. **[^](#cite_ref-93)** Merrill, Ronald T.; McElhinny, Michael W.; McFadden, Phillip L. (1996). "2. The present geomagnetic field: analysis and description from historical observations". *The magnetic field of the earth: paleomagnetism, the core, and the deep mantle*. [Academic Press](/source/Academic_Press). [ISBN](/source/ISBN_(identifier)) [978-0-12-491246-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-491246-5).

1. **[^](#cite_ref-94)** Phillips, Tony (29 December 2003). ["Earth's Inconstant Magnetic Field"](https://web.archive.org/web/20221101165248/https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/). *Science@Nasa*. Archived from [the original](https://science.nasa.gov/science-news/science-at-nasa/2003/29dec_magneticfield/) on 1 November 2022. Retrieved 27 December 2009.

1. **[^](#cite_ref-FOOTNOTEIda2021467Eq._9.99_95-0)** [Ida 2021](#CITEREFIda2021), p. 467, Eq. 9.99.

1. **[^](#cite_ref-96)** Rote, Donald M. (2004). "Magnetic Levitation". *Encyclopedia of Energy*: 691–703. [doi](/source/Doi_(identifier)):[10.1016/B0-12-176480-X/00182-0](https://doi.org/10.1016%2FB0-12-176480-X%2F00182-0). [ISBN](/source/ISBN_(identifier)) [978-0-12-176480-7](https://en.wikipedia.org/wiki/Special:BookSources/978-0-12-176480-7).

1. **[^](#cite_ref-97)** Boyko, B.A.; Bykov, A.I.; Dolotenko, M.I.; Kolokolchikov, N.P.; Markevtsev, I.M.; Tatsenko, O.M.; Shuvalov, K. (1999). "With record magnetic fields to the 21st Century". *Digest of Technical Papers. 12th IEEE International Pulsed Power Conference. (Cat. No.99CH36358)*. Vol. 2. pp. 746–749. [doi](/source/Doi_(identifier)):[10.1109/PPC.1999.823621](https://doi.org/10.1109%2FPPC.1999.823621). [ISBN](/source/ISBN_(identifier)) [0-7803-5498-2](https://en.wikipedia.org/wiki/Special:BookSources/0-7803-5498-2). [S2CID](/source/S2CID_(identifier)) [42588549](https://api.semanticscholar.org/CorpusID:42588549).

1. **[^](#cite_ref-98)** Daley, Jason. ["Watch the Strongest Indoor Magnetic Field Blast Doors of Tokyo Lab Wide Open"](https://www.smithsonianmag.com/smart-news/strongest-indoor-magnetic-field-blows-doors-tokyo-lab-180970436/). *Smithsonian Magazine*. Retrieved 8 September 2020.

1. **[^](#cite_ref-99)** Tuchin, Kirill (2013). ["Particle production in strong electromagnetic fields in relativistic heavy-ion collisions"](https://doi.org/10.1155%2F2013%2F490495). *Adv. High Energy Phys*. **2013** 490495. [arXiv](/source/ArXiv_(identifier)):[1301.0099](https://arxiv.org/abs/1301.0099). [Bibcode](/source/Bibcode_(identifier)):[2013arXiv1301.0099T](https://ui.adsabs.harvard.edu/abs/2013arXiv1301.0099T). [doi](/source/Doi_(identifier)):[10.1155/2013/490495](https://doi.org/10.1155%2F2013%2F490495). [S2CID](/source/S2CID_(identifier)) [4877952](https://api.semanticscholar.org/CorpusID:4877952).

1. **[^](#cite_ref-100)** Bzdak, Adam; Skokov, Vladimir (29 March 2012). "Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions". *Physics Letters B*. **710** (1): 171–174. [arXiv](/source/ArXiv_(identifier)):[1111.1949](https://arxiv.org/abs/1111.1949). [Bibcode](/source/Bibcode_(identifier)):[2012PhLB..710..171B](https://ui.adsabs.harvard.edu/abs/2012PhLB..710..171B). [doi](/source/Doi_(identifier)):[10.1016/j.physletb.2012.02.065](https://doi.org/10.1016%2Fj.physletb.2012.02.065). [S2CID](/source/S2CID_(identifier)) [118462584](https://api.semanticscholar.org/CorpusID:118462584).

1. **[^](#cite_ref-101)** Kouveliotou, C.; Duncan, R. C.; Thompson, C. (February 2003). "[Magnetars](http://solomon.as.utexas.edu/~duncan/sciam.pdf) [Archived](https://web.archive.org/web/20070611144829/http://solomon.as.utexas.edu/~duncan/sciam.pdf) 11 June 2007 at the [Wayback Machine](/source/Wayback_Machine)". *[Scientific American](/source/Scientific_American)*; Page 36.

1. **[^](#cite_ref-102)** Chapman, Allan (2007). "Peregrinus, Petrus (Flourished 1269)". *Encyclopedia of Geomagnetism and Paleomagnetism*. Dordrecht: Springer. pp. 808–809. [doi](/source/Doi_(identifier)):[10.1007/978-1-4020-4423-6_261](https://doi.org/10.1007%2F978-1-4020-4423-6_261). [ISBN](/source/ISBN_(identifier)) [978-1-4020-3992-8](https://en.wikipedia.org/wiki/Special:BookSources/978-1-4020-3992-8).

1. ^ [***a***](#cite_ref-Whittaker1910_104-0) [***b***](#cite_ref-Whittaker1910_104-1) [***c***](#cite_ref-Whittaker1910_104-2) [***d***](#cite_ref-Whittaker1910_104-3) [***e***](#cite_ref-Whittaker1910_104-4) [***f***](#cite_ref-Whittaker1910_104-5) [***g***](#cite_ref-Whittaker1910_104-6) [***h***](#cite_ref-Whittaker1910_104-7) [***i***](#cite_ref-Whittaker1910_104-8) [***j***](#cite_ref-Whittaker1910_104-9) [***k***](#cite_ref-Whittaker1910_104-10) [***l***](#cite_ref-Whittaker1910_104-11) [***m***](#cite_ref-Whittaker1910_104-12) [***n***](#cite_ref-Whittaker1910_104-13) [Whittaker, E. T.](/source/E._T._Whittaker) (1910). *[A History of the Theories of Aether and Electricity](/source/A_History_of_the_Theories_of_Aether_and_Electricity)*. [Dover Publications](/source/Dover_Publications). [ISBN](/source/ISBN_(identifier)) [978-0-486-26126-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-486-26126-3). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

1. **[^](#cite_ref-106)** Williams, L. Pearce (1974). ["Oersted, Hans Christian"](https://archive.org/details/dictionaryofscie10gill/page/184). In Gillespie, C. C. (ed.). *Dictionary of Scientific Biography*. New York: Charles Scribner's Sons. p. 185.

1. **[^](#cite_ref-107)** Blundell, Stephen J. (2012). *Magnetism: A Very Short Introduction*. OUP Oxford. p. 31. [ISBN](/source/ISBN_(identifier)) [978-0-19-163372-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-19-163372-0).

1. ^ [***a***](#cite_ref-Tricker23_108-0) [***b***](#cite_ref-Tricker23_108-1) Tricker, R. A. R. (1965). [*Early electrodynamics*](https://archive.org/details/earlyelectrodyna0000tric). Oxford: Pergamon. p. [23](https://archive.org/details/earlyelectrodyna0000tric/page/23).

1. **[^](#cite_ref-109)** Erlichson, Herman (1998). ["The experiments of Biot and Savart concerning the force exerted by a current on a magnetic needle"](https://doi.org/10.1119%2F1.18878). *American Journal of Physics*. **66** (5): 389. [Bibcode](/source/Bibcode_(identifier)):[1998AmJPh..66..385E](https://ui.adsabs.harvard.edu/abs/1998AmJPh..66..385E). [doi](/source/Doi_(identifier)):[10.1119/1.18878](https://doi.org/10.1119%2F1.18878).

1. **[^](#cite_ref-110)** Frankel, Eugene (1972). *Jean-Baptiste Biot: The career of a physicist in nineteenth-century France*. Princeton University: Doctoral dissertation. p. 334.

1. **[^](#cite_ref-112)** [Lord Kelvin of Largs](https://www.physik.uni-augsburg.de/lehrstuehle/did/Physik/Schulservice/Physikerkalender/06_26_Lord-Kelvin-of-Largs.pdf). physik.uni-augsburg.de. 26 June 1824

1. **[^](#cite_ref-h202_113-0)** Huurdeman, Anton A. (2003) *The Worldwide History of Telecommunications*. Wiley. [ISBN](/source/ISBN_(identifier)) [0-471-20505-2](https://en.wikipedia.org/wiki/Special:BookSources/0-471-20505-2). p. 202

1. **[^](#cite_ref-114)** ["The most important Experiments – The most important Experiments and their Publication between 1886 and 1889"](https://www.hhi.fraunhofer.de/fraunhofer-hhi-the-institute/about-us/history-of-hhi/the-most-important-experiments.html). Fraunhofer Heinrich Hertz Institute. Retrieved 19 February 2016.

1. **[^](#cite_ref-115)** [*Networks of Power: Electrification in Western Society, 1880–1930*](https://books.google.com/books?id=g07Q9M4agp4C&pg=PA117). JHU Press. March 1993. p. 117. [ISBN](/source/ISBN_(identifier)) [978-0-8018-4614-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8018-4614-4).

1. **[^](#cite_ref-116)** Thomas Parke Hughes, *Networks of Power: Electrification in Western Society, 1880–1930*, pp. 115–118

1. **[^](#cite_ref-117)** Ltd, Nmsi Trading; Smithsonian Institution (1998). [*Robert Bud, Instruments of Science: An Historical Encyclopedia*](https://books.google.com/books?id=1AsFdUxOwu8C&pg=PA204). Taylor & Francis. p. 204. [ISBN](/source/ISBN_(identifier)) [978-0-8153-1561-2](https://en.wikipedia.org/wiki/Special:BookSources/978-0-8153-1561-2). Retrieved 18 March 2013.

1. **[^](#cite_ref-118)** [U.S. patent 381,968](https://patents.google.com/patent/US381968)

1. **[^](#cite_ref-119)** Porter, H. F. J.; Prout, Henry G. (January 1924). "A Life of George Westinghouse". *The American Historical Review*. **29** (2): 129. [doi](/source/Doi_(identifier)):[10.2307/1838546](https://doi.org/10.2307%2F1838546). [hdl](/source/Hdl_(identifier)):[2027/coo1.ark:/13960/t15m6rz0r](https://hdl.handle.net/2027%2Fcoo1.ark%3A%2F13960%2Ft15m6rz0r). [ISSN](/source/ISSN_(identifier)) [0002-8762](https://search.worldcat.org/issn/0002-8762). [JSTOR](/source/JSTOR_(identifier)) [1838546](https://www.jstor.org/stable/1838546).

1. **[^](#cite_ref-120)** ["Galileo Ferraris (March 1888) *Rotazioni elettrodinamiche prodotte per mezzo di correnti alternate* (Electrodynamic rotations by means of alternating currents), memory read at Accademia delle Scienze, Torino, in *Opere di Galileo Ferraris*, Hoepli, Milano, 1902 vol I pages 333 to 348"](https://web.archive.org/web/20210709182151/http://ieeemilestones.ethw.org/images/a/a0/1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf) (PDF). Archived from [the original](http://ieeemilestones.ethw.org/images/a/a0/1-Galileo_Ferraris_Rotating-field--Rotazioni_Elettrodinamiche.pdf) (PDF) on 9 July 2021. Retrieved 2 July 2021.

1. **[^](#cite_ref-:0_121-0)** Feynman, R. P. (1950). ["Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction"](https://web.archive.org/web/20200914231627/https://authors.library.caltech.edu/3528/). *Physical Review*. **80** (3): 440–457. [Bibcode](/source/Bibcode_(identifier)):[1950PhRv...80..440F](https://ui.adsabs.harvard.edu/abs/1950PhRv...80..440F). [doi](/source/Doi_(identifier)):[10.1103/PhysRev.80.440](https://doi.org/10.1103%2FPhysRev.80.440). Archived from [the original](https://authors.library.caltech.edu/3528/) on 14 September 2020. Retrieved 23 September 2019.

### Sources

- [Griffiths, David J.](/source/David_J._Griffiths) (1999). *[Introduction to Electrodynamics](/source/Introduction_to_Electrodynamics)* (3rd ed.). Pearson. [ISBN](/source/ISBN_(identifier)) [0-13-805326-X](https://en.wikipedia.org/wiki/Special:BookSources/0-13-805326-X).

- [Griffiths, David J.](/source/David_J._Griffiths) (2017). *[Introduction to Electrodynamics](/source/Introduction_to_Electrodynamics)* (4th ed.). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-1-108-35714-2](https://en.wikipedia.org/wiki/Special:BookSources/978-1-108-35714-2).

- Ida, Nathan (2021). *Engineering Electromagnetics* (4th ed.). Springer. [ISBN](/source/ISBN_(identifier)) [978-3-030-15556-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-030-15556-8).

- [Jackson, John David](/source/John_David_Jackson_(physicist)) (1975). [*Classical electrodynamics*](https://archive.org/details/classicalelectro00jack_0) (2nd ed.). New York: Wiley. [ISBN](/source/ISBN_(identifier)) [978-0-471-43132-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-43132-9).

- [Jackson, John David](/source/John_David_Jackson_(physicist)) (1998). *Classical electrodynamics* (3rd ed.). New York: Wiley. [ISBN](/source/ISBN_(identifier)) [0-471-30932-X](https://en.wikipedia.org/wiki/Special:BookSources/0-471-30932-X).

- Ling, Samuel J.; Moebs, William; Sanny, Jeff (2016). *University Physics Volume 2*. OpenStax.

- Purcell, E. (2011). [*Electricity and Magnetism*](https://archive.org/details/electricitymagne00purc_621) (2nd ed.). Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-1-107-01360-5](https://en.wikipedia.org/wiki/Special:BookSources/978-1-107-01360-5).

- Purcell, Edward M.; Morin, David J. (2013). [*Electricity and Magnetism*](https://www.cambridge.org/core/product/identifier/9781139012973/type/book). Cambridge University Press. [doi](/source/Doi_(identifier)):[10.1017/cbo9781139012973](https://doi.org/10.1017%2Fcbo9781139012973). [ISBN](/source/ISBN_(identifier)) [978-1-139-01297-3](https://en.wikipedia.org/wiki/Special:BookSources/978-1-139-01297-3).

- Rothwell, E. J.; Cloud, M. J. (2010). [*Electromagnetics*](https://books.google.com/books?id=7AHLBQAAQBAJ). Taylor & Francis. [ISBN](/source/ISBN_(identifier)) [1-4200-5826-6](https://en.wikipedia.org/wiki/Special:BookSources/1-4200-5826-6).

- Stratton, Julius Adams (1941). *Electromagnetic Theory* (1st ed.). McGraw-Hill. [ISBN](/source/ISBN_(identifier)) [978-0-07-062150-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-07-062150-3). {{[cite book](https://en.wikipedia.org/wiki/Template:Cite_book)}}: ISBN / Date incompatibility ([help](https://en.wikipedia.org/wiki/Help:CS1_errors#invalid_isbn_date))

- Wangsness, Roald K. (1986). *Electromagnetic Fields* (2nd ed.). Hamilton Printing Company. [ISBN](/source/ISBN_(identifier)) [0-471-81186-6](https://en.wikipedia.org/wiki/Special:BookSources/0-471-81186-6).

### Further reading

- Jiles, David (1994). *Introduction to Electronic Properties of Materials* (1st ed.). Springer. [ISBN](/source/ISBN_(identifier)) [978-0-412-49580-9](https://en.wikipedia.org/wiki/Special:BookSources/978-0-412-49580-9).

- Tipler, Paul (2004). *Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.)*. W. H. Freeman. [ISBN](/source/ISBN_(identifier)) [978-0-7167-0810-0](https://en.wikipedia.org/wiki/Special:BookSources/978-0-7167-0810-0). [OCLC](/source/OCLC_(identifier)) [51095685](https://search.worldcat.org/oclc/51095685).

### External links

- Media related to [Magnetic fields](https://commons.wikimedia.org/wiki/Category:Magnetic_fields) at Wikimedia Commons.

- Crowell, B., "*[Electromagnetism](http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html) [Archived](https://web.archive.org/web/20100430152732/http://www.lightandmatter.com/html_books/0sn/ch11/ch11.html) 30 April 2010 at the [Wayback Machine](/source/Wayback_Machine)*".

- Nave, R., "*[Magnetic Field](http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magfie.html)*". HyperPhysics.

- "*Magnetism*", [The Magnetic Field](https://web.archive.org/web/20060709195126/http://theory.uwinnipeg.ca/physics/mag/node2.html#SECTION00110000000000000000) (archived 9 July 2006). theory.uwinnipeg.ca.

- Hoadley, Rick, "*[What do magnetic fields look like](http://my.execpc.com/~rhoadley/magfield.htm) [Archived](https://web.archive.org/web/20110219102314/http://my.execpc.com/~rhoadley/magfield.htm) 19 February 2011 at the [Wayback Machine](/source/Wayback_Machine)?*" 17 July 2005.

v t e Magnetism Magnetic response diamagnetism superdiamagnetism paramagnetism superparamagnetism Van Vleck paramagnetism Magnetic states altermagnetism antiferromagnetism ferrimagnetism ferromagnetism superferromagnetism ferromagnetic superconductor helimagnetism metamagnetism mictomagnetism spin glass amorphous magnetism spin ice

Authority control databases International GND National United States France BnF data Japan Czech Republic Israel Other Encyclopedia of Modern Ukraine Yale LUX

---
Adapted from the Wikipedia article [Magnetic field](https://en.wikipedia.org/wiki/Magnetic_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Magnetic_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.