# Force density

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{{Infobox physical quantity
| name = Force density
| width =
| background =
| image = 
| caption = Forces density is force per unit volume.
| unit = N·m<sup>−3</sup>
| otherunits = 
| symbols = <math>\mathbf{f}</math>
| baseunits = [kg](/source/kilogram)·[m](/source/metre)<sup>−2</sup>·[s](/source/second)<sup>−2</sup>
| dimension = wikidata
| derivations = 
}}

{{Classical mechanics|expanded=Fundamental concepts}}

In [fluid mechanics](/source/fluid_mechanics), the '''force density''' is the negative [gradient of pressure](/source/gradient_of_pressure). It has the physical dimensions of [force](/source/force) per unit [volume](/source/volume). Force density is a [vector field](/source/vector_field) representing the [flux](/source/flux) density of the [hydrostatic force](/source/hydrostatic_force) within the bulk of a [fluid](/source/fluid).  Force density is represented by the symbol '''f''',<ref>[http://scienceworld.wolfram.com/physics/ForceDensity.html Force Density]. Eric Weisstein's World of Physics. Accessed March 8th, 2012.</ref> and given by the following equation, where ''p'' is the [pressure](/source/pressure):

:<math>\mathbf{f} = - \nabla p </math>.

The net force on a differential [volume element](/source/volume_element) ''dV'' of the fluid is:

:<math>d\mathbf{F} = \mathbf{f}dV</math>

Force density acts in different ways which is caused by the boundary conditions. There are stick-slip boundary conditions and stick boundary conditions which affect force density.

In a sphere placed in an arbitrary non-stationary flow field of viscous incompressible fluid for stick boundary conditions where the force density's calculations leads to show the generalisation of [Faxen's theorem](/source/Fax%C3%A9n's_law) to force multipole moments of arbitrary order.

In a sphere moving in an incompressible fluid in a non-stationary flow with mixed stick-slip boundary condition where the force of density shows an expression of the Faxén type for the total force, but the total torque and the symmetric force-dipole moment.<ref>Physica A: Statistical Mechanics and its Applications Volume 84, Issue 3, Pages 435-641 (1976) Accessed 19 January 2015</ref>

The force density at a point in a fluid, divided by the [density](/source/density), is the [acceleration](/source/acceleration) of the fluid at that point.

The force density '''f''' is defined as the force per unit volume, so that the net force can be calculated by:
:<math>\mathbf{F}=\int f(\mathbf{r})d^3 \mathbf{r} </math>.

The force density in an electromagnetic field is given in CGS by:
:<math>\mathbf{f}=\rho \mathbf{E}+ \frac{\mathbf{J}}{c} \times \mathbf{B} </math>,

where <math>\rho </math> is the charge density, '''E''' is the electric field, '''J''' is the current density, c is the speed of light, and '''B''' is the magnetic field.<ref>[http://scienceworld.wolfram.com/physics/ForceDensity.html Force Density]. Eric Weisstein's World of Physics. Accessed 17 January 2015.</ref>

==See also==
* [Body force](/source/Body_force)
* [Pressure gradient](/source/Pressure_gradient)
* [Gradient](/source/Gradient)

==References==
<references/>

{{DEFAULTSORT:Force Density}}
Category:Density

{{Fluiddynamics-stub}}

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