Minimum deviation explained

In a prism, the angle of deviation decreases with increase in the angle of incidence 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,, or).

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

n _{21} = \dfrac{\sin \left(\dfrac{A + D _{m} }\right)}

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

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.^[1] Also, the angles of refractions are equal i.e. . The angle of incidence and angle of emergence equal each other . 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.

From the angle sum of $\backslash triangle\; OPQ$,

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

Using the exterior angle theorem in $\backslash triangle\; PQR$,

D _{m} = \angle RPQ + \angle RQP \implies D _{m} = i - r + i - r \implies 2 r + D _{m} = 2 i \implies A + D _{m} = 2 i \implies i = (A + D _{m})/(2)

This can also be derived by putting in the prism formula:

From Snell's law,

n _{21} = \dfrac{\sin i}{\sin r}

$$\backslash therefore\; n\_\; =\; \backslash dfrac$$^{[2] [3]}

$$\backslash therefore\; D\_m\; =\; 2\; \backslash sin^\; \backslash left(n\; \backslash sin\; \backslash left(\backslash frac\backslash right)\backslash right)\; -\; A$$

(where is the refractive index, is the Angle of Prism and 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.^{[4] [5]}

Worked out examples:

Answer: 37°, 49°

Solution:

Here,,

Plugging them in the above formula,

$\backslash frac=\; 1.5$

$\backslash implies\; \backslash frac=\; 1.5$

$\backslash implies\; \backslash sin\; \backslash left(30\; +\; \backslash frac\; \backslash right)\; =\; 1.5\; \backslash times\; 0.5$

$\backslash implies\; 30\; +\; \backslash frac\; =\; \backslash sin^(0.75)$

$\backslash implies\; \backslash frac\; =\; 48.6\; -\; 30$

$\backslash implies\; \backslash delta\; =\; 2\; \backslash times\; 18.6$

$\backslash therefore\; \backslash delta\; \backslash approx\; 37^\backslash circ$

Also,

$i\; =\; \backslash frac\; =\; \backslash frac\; \backslash approx\; 49^\backslash circ$

This is also apparent in the graph below.

Answer: 60°

Solution:

Here,$\backslash delta\; =\; r$

$\backslash implies\; \backslash delta\; =\; \backslash frac$

Using the above formula,

$\backslash frac=\; 1.4$

$\backslash implies\; \backslash frac=\; \backslash frac$

$\backslash implies\; \backslash frac=\; \backslash frac$

$\backslash therefore\; A\; =\; 60^\backslash circ$

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 in the prism formula using Snell's law:

$$\backslash delta\; =\; i\; -\; A\; +\; \backslash sin^\; \backslash left(n\; \backslash cdot\; \backslash sin\backslash left(A\; -\; \backslash sin^\backslash left(\backslash frac\backslash right)\backslash right)\backslash right)=f(i)(say)$$

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

Putting

f'(i) =0 , we get,(\cos\left(A - \sin ^{-1} \left( (\sin i)/(u) \right)\right)\cos i)/(\sqrt{\left( 1- u ^{2} \sin ^{2} \left(A - \sin ^{-1} \left( (\sin i)/(u) \right)\right)\right)\left( 1- (\sin ^{2} i)/(u ^{2}) \right)) }=1, 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 here

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 and are very small,

\begin{align} n & ≈ \dfrac{ (A + D _{m})/(2) }\\n & = \frac\\D_m & = An - A\end

$$\backslash therefore\; D\_\; =\; A(n\; -\; 1)$$

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, and are small,

n ≈ (i)/(r _{1}) , n ≈ (e)/(r _{2})

From the prism formula,

\begin{align} \delta & = n r _{1} + n r _{2} - A \\ & = n(r _{1} + r _{2)} - A \\ & = nA - A \\ & = A(n - 1 ) \end{align}

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

Experimental determination

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.^{[6] [7]}

Minimum angle of dispersion

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.

For a thin prism, the deviation of violet light,

\delta _{v} is (n _{v-1)A} and that of red light, \delta _{r} is (n _{r-1)A} . The difference in the deviation between red and violet light, (\delta _{v-\delta} _{r)=(n} _{v-n} _{r)A} is called the Angular Dispersion produced by the prism.

Applications

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°).^[8]

It is also responsible for phenomena like halos and 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°.^[9]

See also

• Prism
• Refraction
• Geometrical optics

References

1. Web site: Optics-Prism. A-Level Physics Tutor.
2. Web site: Refraction through Prisms. SchoolPhysics.
3. Web site: Prism. HyperPhysics.
4. Web site: Minimum Deviation by a Prism. Mark A. Peterson. mtholyoke. Mount Holyoke College. dead. https://web.archive.org/web/20190523072850/http://www.mtholyoke.edu/~mpeterso/classes/phys103/geomopti/MinDev.html. 2019-05-23.
5. Book: Physics Part II Textbook for Class IX. NCERT. 331. en. Chapter Nine, RAY OPTICS AND OPTICAL INSTRUMENTS.
6. Web site: Angle of Minimum Deviation. Scribd. en.
7. Web site: Experimental set up for the measurements of angle of minimum deviation using prism spectrometer. ResearchGate. en.
8. Web site: Rainbow. www.schoolphysics.co.uk.
9. Web site: Halo 22°. HyperPhysics.

External links

• Minimum Deviation Part 1 and Part 2 at Khan Academy
• Refraction through a Prism in NCERT Textbook
• Minimum Deviation by Prism by Mark A Peterson, Mount Holyoke College