Minimum deviation

{{short description|Condition when the angle of deviation is minimal in a prism}}

In a [[Prism (optics)|prism]], the angle of deviation ({{mvar|δ}}) decreases with increase in the angle of incidence ({{mvar|i}}) up to a particular angle. This angle of incidence where the angle of deviation in a prism is minimum is called the '''minimum deviation position''' of the prism and that very deviation angle is known as the '''minimum angle of deviation''' (denoted by {{math|''δ''min}}, {{mvar|Dλ}}, or {{mvar|Dm}}).

[[File:Deviation angle of a beam of light entering a medium with refractive index.png|thumb|Light is deflected as it enters a material with refractive index > 1.]] [[File:Deflection of light ray passing through prism.png|thumb|A ray of light is deflected twice in a prism. The sum of these deflections is the deviation angle.]] [[File:Minimum deflection of light ray passing through prism.png|thumb|When the entrance and exit angles are equal, the deviation angle of a ray passing through a prism will be minimal.]]

The angle of minimum deviation is related with the [[refractive index]] as:

<math> n_{21} = \dfrac{\sin \left(\dfrac{A + D_{m}}{2}\right)}{\sin \left(\dfrac{A}{2}\right)} </math>

This is useful to calculate the refractive index of a material. Rainbow and halo occur at minimum deviation. Also, a thin prism is always set at minimum deviation.

==Formula==

{{Expand Section|the derivation of the expression for minimum deviation using Calculus|date=June 2020}}

In minimum deviation, the refracted ray in the prism is parallel to its base. In other words, the light ray is symmetrical about the axis of symmetry of the prism.<ref name=":1" /><ref name=":2">{{Cite web|url=https://www.a-levelphysicstutor.com/optics-prisms.php|title=Optics-Prism|website=A-Level Physics Tutor}}</ref><ref name=":0" /> Also, the angles of refractions are equal i.e. {{math|1=''r''1 = ''r''2}}. [[Angle of incidence (optics)|The angle of incidence]] and angle of emergence equal each other ({{math|1=''i'' = ''e''}}). This is clearly visible in the graph below.

The formula for minimum deviation can be derived by exploiting the geometry in the prism. The approach involves replacing the variables in the [[Snell's law]] in terms of the Deviation and Prism Angles by making the use of the above properties.

[[File:Minimum Deviation.jpg]]

From the [[Sum of angles of a triangle|angle sum]] of <math display="inline">\triangle OPQ</math>,

<math>A + \angle OPQ + \angle OQP = 180^\circ</math> <math>\implies A = 180^\circ - (90 - r) - (90 - r)</math> <math>\implies r = \frac{A}{2}</math>

Using the [[exterior angle theorem]] in <math display="inline"> \triangle PQR</math>,

<math> D_{m} = \angle RPQ + \angle RQP </math> <math> \implies D_{m} = i - r + i - r </math> <math> \implies 2r + D_{m}= 2i </math> <math> \implies A + D_{m} = 2i </math> <math> \implies i = \frac{A + D_{m}} {2} </math>

This can also be derived by putting {{math|1=''i'' = ''e''}} in the [[prism formula]]: {{math|1=''i'' + ''e'' = ''A'' + ''δ''}}

From [[Snell's law]],

<math>n_{21} = \dfrac{\sin i}{\sin r}</math>

<math display="block"> \therefore n_{21} = \dfrac{\sin \left(\dfrac{A + D_{m}}{2}\right)}{\sin \left(\dfrac{A}{2}\right)} </math><ref name=":3">{{Cite web|title=Refraction through Prisms|url=http://www.schoolphysics.co.uk/age16-19/Optics/Refraction/text/Prisms_/index.html|website=SchoolPhysics}}</ref><ref name=":0" /><ref name=":1" /><ref name=":2" /><ref name=":4">{{Cite web|title=Prism|url=http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/prism.html#c2|website=HyperPhysics}}</ref>{{Excessive citations inline|date=June 2020}}

<math display="block">\therefore D_m = 2 \sin^{-1} \left(n \sin \left(\frac{A}{2}\right)\right) - A </math>

(where {{mvar|n}} is the refractive index, {{mvar|A}} is the Angle of Prism and {{mvar|Dm}} is the Minimum Angle of Deviation.)

This is a convenient way used to measure the refractive index of a material(liquid or gas) by directing a light ray through a prism of negligible thickness at minimum deviation filled with the material or in a glass prism dipped in it.<ref name=":4" /><ref name=":0">{{cite web|url=https://www.mtholyoke.edu/~mpeterso/classes/phys103/geomopti/MinDev.html|title=Minimum Deviation by a Prism|last=Mark A. Peterson|website=mtholyoke|publisher=[[Mount Holyoke College]]|url-status=dead|archive-url=https://web.archive.org/web/20190523072850/http://www.mtholyoke.edu/~mpeterso/classes/phys103/geomopti/MinDev.html|archive-date=2019-05-23}}</ref><ref name=":1">{{Cite book|url=http://ncert.nic.in/NCERTS/l/leph201.pdf|title=Physics Part II Textbook for Class IX|publisher=NCERT|pages=331|language=en|chapter=Chapter Nine, RAY OPTICS AND OPTICAL INSTRUMENTS}}</ref>

Worked out examples:

{{collapse top|The refractive index of glass is 1.5. The minimum angle of deviation for an equilateral prism along with the corresponding angle of incidence is desired.}}

Answer: 37°, 49°

Solution:

Here, {{math|1=''A'' = 60°}}, {{math|1=''n'' = 1.5}}

Plugging them in the above formula,

<math display="inline"> \frac{\sin \left(\frac{60 + \delta}{2} \right)}{\sin \left(\frac{60}{2} \right)}= 1.5 </math>

<math display="inline"> \implies \frac{\sin \left(30 + \frac{\delta}{2} \right)}{\sin(30)}= 1.5 </math>

<math display="inline"> \implies \sin \left(30 + \frac{\delta}{2} \right) = 1.5 \times 0.5 </math>

<math display="inline"> \implies 30 + \frac{\delta}{2} = \sin^{-1}(0.75) </math>

<math display="inline"> \implies \frac{\delta}{2} = 48.6 - 30 </math>

<math display="inline"> \implies \delta = 2 \times 18.6 </math>

<math display="inline"> \therefore \delta \approx 37^\circ</math>

Also,

<math display="inline"> i = \frac{(A + \delta)}{2} = \frac{60 + 2 \times 18.6}{2} \approx 49^\circ</math>

This is also apparent in the graph below.

{{collapse top|If the minimum angle of deviation of a prism of refractive index 1.4 equals its refracting angle, the angle of the prism is desired.}}

Answer: 60°

Solution:

Here, <math display="inline"> \delta = r </math>

<math display="inline">\implies \delta = \frac{A}{2} </math>

Using the above formula,

<math display="inline"> \frac{\sin \left(\frac{A + \frac{A}{2}}{2}\right)}{\sin \left(\frac{A}{2} \right)}= 1.4 </math>

<math display="inline"> \implies \frac{\sin \left(\frac{3A}{4} \right)}{\sin \left(\frac{A}{2} \right)}= \frac{\frac{1}{2}}{ \frac{1}{\sqrt 2}} </math>

<math display="inline"> \implies \frac{\sin \left(\frac{3A}{4} \right)}{\sin \left(\frac{A}{2} \right)}= \frac{\sin 45^\circ}{\sin 30^\circ}</math>

<math display="inline">\therefore A = 60^\circ </math>

Also, the variation of the angle of deviation with an arbitrary angle of incidence can be encapsulated into a single equation by expressing δ in terms of {{mvar|i}} in the prism formula using Snell's law:

<math display="block">\delta = i - A + \sin^{-1} \left(n \cdot \sin\left(A - \sin^{-1}\left(\frac{\sin i}{n}\right)\right)\right)=f(i)</math>

Finding the minima of this equation will also give the same relation for minimum deviation as above.

Putting <math>f'(i)=0</math>, we get,

<math>\frac{\cos\left(A-\sin^{-1}\left(\frac{\sin i}{u}\right)\right)\cos i}{\sqrt{\left(1-u^{2}\sin^{2}\left(A-\sin^{-1}\left(\frac{\sin i}{u}\right)\right)\right)\left(1-\frac{\sin^{2}i}{u^{2}}\right)}}=1</math>, and by solving this equation we can obtain the value of angle of incidence for a definite value of angle of prism and the value of relative refractive index of the prism for which the minimum angle of deviation will be obtained. The equation and description are given [https://www.desmos.com/calculator/0jkerecdx3 here]

[[File:Minimum deviation.png|thumb|center|500px|In this graph of the angle of deviation vs the angle of incidence, δ corresponds to two values of i and e(i'). For minimum deviation, however, i equals e.]]

===For thin prism===

In a thin or small angle prism, as the angles become very small, the [[sine]] of the angle nearly equals the angle itself and this yields many useful results.

Because {{mvar|Dm}} and {{mvar|A}} are very small,

<math>

\begin{align} n & \approx \dfrac{\frac{A + D_{m}}{2}}{\frac{A}{2}}\\ n & = \frac{A + D_m}{A}\\ D_m & = An - A \end{align} </math>

<math display="block"> \therefore D_{m} = A(n - 1) </math><ref name=":1" /><ref name=":3" />

Using a similar approach with the [[Snell's law]] and the [[prism formula]] for an in general thin-prism ends up in the very same result for the deviation angle.

Because {{mvar|i}}, {{mvar|e}} and {{mvar|r}} are small,

<math> n \approx \frac{i}{r_1}, n \approx \frac{e}{r_2} </math>

From the prism formula, <math> \begin{align} \delta & = n r_1 + n r_2 - A \\ & = n(r_1 + r_2) - A \\ & = nA - A \\ & = A(n - 1) \end{align} </math>

Thus, it can be said that a thin prism is always in minimum deviation.

==Experimental determination==

{{Further|Prism spectrometer}} {{Expand Section|date=May 2020|virtual simulation, video, detailed explanation, etc}}

Minimum deviation can be found manually or with spectrometer. Either the prism is kept fixed and the incidence angle is adjusted or the prism is rotated keeping the light source fixed.<ref>{{Cite web|title=Angle of Minimum Deviation|url=https://www.scribd.com/document/201177761/Angle-of-Minimum-Deviation|website=Scribd|language=en}}</ref><ref>{{Cite web|title=Experimental set up for the measurements of angle of minimum deviation using prism spectrometer|url=https://www.researchgate.net/figure/Experimental-set-up-for-the-measurements-of-angle-of-minimum-deviation-using-prism_fig1_326377418|website=ResearchGate|language=en}}</ref><ref>{{Cite web |title=Theory of the Prism Spectrometer - Experiment {{!}} UKEssays.com |url=https://www.ukessays.com/essays/physics/theory-prism-spectrometer-experiment-7823.php |access-date=2025-03-11 |website=www.ukessays.com |language=en}}</ref>

[[File:Experiment setup.svg]]

==Minimum angle of dispersion== [[File:Prisms-disp-diag02.jpg|thumb|The angle of dispersion in a prism]] {{Expand Section||date=May 2020}}

The minimum angle of dispersion for white light is the difference in minimum deviation angle between red and violet rays of a light ray through a prism.<ref name=":2" />

For a thin prism, the deviation of violet light, <math>\delta_v</math> is <math>(n_v-1)A</math> and that of red light, <math>\delta_r</math> is <math>(n_r-1)A</math>. The difference in the deviation between red and violet light, <math>(\delta_v-\delta_r)=(n_v-n_r)A</math> is called the Angular Dispersion produced by the prism.

==Applications== {{Further|Rainbow|Sun dog}}

[[File:Rainbow1.svg|thumb|Drawing radii to the points of interference reveals that the angles of refraction are equal, thereby proving minimum deviation.]]

One of the factors that causes a rainbow is the bunching of light rays at the minimum deviation angle that is close to the rainbow angle (42°).<ref name=":0" /><ref>{{Cite web|title=Rainbow|url=http://www.schoolphysics.co.uk/age16-19/Optics/Refraction/text/Rainbow_/index.html|website=www.schoolphysics.co.uk}}</ref>

[[File:22_Degree_halo_Rotating_Hexagonal_Ice_Crystal_Complete_cycle.gif|thumb]]

It is also responsible for phenomena like [[Halo (optical phenomenon)|halos]] and [[Sun dog|sundogs]], produced by the deviation of sunlight in mini prisms of hexagonal ice crystals in the air bending light with a minimum deviation of 22°.<ref name=":0" /><ref>{{Cite web|title=Halo 22°|url=http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/halo22.html|website=HyperPhysics}}</ref>

==See also== {{Portal|physics}}

* [[Prism (optics)|Prism]] * [[Refraction]] * [[Geometrical optics]]

==References== <references/>

==External links== * Minimum Deviation [https://www.khanacademy.org/science/in-in-class-12th-physics-india/in-in-ray-optics-and-optical-instruments/in-in-ray-optics-and-optical-instruments/in-in-refraction-in-prisms/v/minimum-deviation-in-prism?modal=1 Part 1] and [https://www.khanacademy.org/science/in-in-class-12th-physics-india/in-in-ray-optics-and-optical-instruments/in-in-ray-optics-and-optical-instruments/in-in-refraction-in-prisms/v/deriving-prism-formula?modal=1 Part 2] at [[Khan Academy]] * [https://s3.ap-south-1.amazonaws.com/aglasem-cdn/aglasem-doc/703f1f62-4d77-11ea-a344-02f21f5619c4/23.jpg Refraction through a Prism] in [[National Council of Educational Research and Training|NCERT Textbook]] * [https://www.mtholyoke.edu/~mpeterso/classes/phys103/geomopti/MinDev.html Minimum Deviation by Prism] {{Webarchive|url=https://web.archive.org/web/20190523072850/http://www.mtholyoke.edu/~mpeterso/classes/phys103/geomopti/MinDev.html |date=2019-05-23 }} by Mark A Peterson, [[Mount Holyoke College]]

[[Category:Geometrical optics]] [[Category:Light]]