Prism

PRISM

A prism has two plane surfaces AB and AC inclined to each other as shown in following figure. $\angle$A is called the angle of prism or refracting angle.

Refraction Through A

A light ray striking at one face of a triangular glass prism gets refracted twice and emerges out from the other face as shown above. The angle between the emergent and the incident rays is called the angle of deviation (D). The angle between the two refracting faces involved is called the refracting angle (A) of the prism.

From AXY, we have: A + (90° – r₁) + (90° – r₂) = 180°

$\Rightarrow r_1+r_2=A$ (i)

Deviation D = (i – r₁) + (e – r₂)

D = (i + e) – (r₁ + r₂)

$\Rightarrow$ D = i + e – A

$\Rightarrow$ A + D = i + e (ii)

We also have two equations from Snell's Law at X & Y.

$\frac{\sin r}{\sin r_1}=\mu And\frac{\sin r_2}{\sin e}=\frac{1}{\mu }$

(i) Angle of Deviation

It can be easily seen that if we reverse the emergent ray, it goes back along the same path. The angles of incidence and emergence get interchanged but the angle of deviation remains same.

Hence the same angle of deviation D is possible for two different angles of incidence :$\theta$₁ and$\theta$₂, where$\theta$₁ +$\theta$₂ = A + D.

(ii) Minimum Deviation

The angle of deviation is minimum when the path of light ray through the prism is symmetrical. i.e., angle of incidence = angle of emergence

i = e

$\Rightarrow r_1=r_2$

and hence $A+D_{\min }=i+i=2i$

also $A=r_1+r_2=2r$

from $\frac{\sin i}{\sin r_1}=\mu ,$ we have:

$\frac{\sin \left(\frac{A+D_{\min }}{2}\right)}{\sin \frac{A}{2}}=\mu$

(iii) Grazing Incidence

When i = 90°, the incident ray grazes along the surface of the prism and the angle of refraction inside the prism becomes equal to the critical angle for glass - air. This is known as grazing incidence.

(iv) Grazing Emergence

When e = 90°, the emergent ray grazes along the prism surface. This happens when the light ray strikes the second face of the prism at the critical angle for glass - air. This is known as grazing emergence.

(v) Maximum Deviation

The angle of deviation is same for both the above cases (grazing incidence & grazing emergence) and it is also the maximum possible deviation if the light ray is to emerge out from the other face without any total internal reflection.

(vi) Dispersion of Light

When a ray of light passes through a prism, it splits up into rays of constituent colours or wavelengths. This phenomenon is called dispersion of light.

The refractive index of a medium is different for light rays of different wavelengths. Larger the wavelengths, the lesser is the refractive index.

i.e., ${\mu }_{red}<{\mu }_{violet}because{\lambda }_{red}>{\lambda }_{violet.}$

A ray of white light passing through a prism gets splits into different colours because the deviation is more for violet and less for red. This phenomenon is known as dispersion.

For a prism with very small refracting angle A, the deviation D is given by:

$d\approx \left(\mu -1\right)A$

Hence deviation of violet = D_v = ($\mu$_v, ­­– 1)A

and for red = D_r = ($\mu$_r – 1)A

The angle between the red and violet rays is known as the angle of dispersion.

The angle of dispersion $=\varphi =D_v-D_r=\left({\mu }_v-{\mu }_r\right)A$

Angle of mean deviation $=D_m=\frac{D_v+D_r}{2}=\left(\frac{{\mu }_v+{\mu }_r}{2}-1\right)A$

Dispersive power of glass $=\omega =\frac{\varphi }{D_m}=\frac{{\mu }_v-{\mu }_r}{\frac{{\mu }_v+{\mu }_r}{2}-1}$

This phenomenon arises due to the fact that refractive index varies with wavelength. It has been observed for a prism that $\mu$ decreases with the increase of wavelength, i.e. ${\mu }_{blue}>{\mu }_{red}$.

Angular dispersion, $\theta ={\delta }_v-{\delta }_r$

(vii) Dispersive power

$\omega =\frac{{\delta }_v-{\delta }_r}{\delta }$where d is deviation of mean ray (yellow)

As ${\delta }_v=\left({\mu }_v-1\right)A,{\delta }_r=\left({\mu }_r-1\right)A,$

$\therefore \omega =\frac{{\mu }_v-{\mu }_r}{\mu -1}$

(viii) Deviation without Dispersion

This mean an achromatic combination of two prisms in which net or resultant dispersion is zero and deviation is produced. For the two prisms,

$\left({\mu }_v-{\mu }_r\right)A+\left(\mu {{}^{\prime }}_v-\mu {{}^{\prime }}_r\right)A^{\prime }=0$

$\Rightarrow A^{\prime }=-\frac{\left({\mu }_v-{\mu }_r\right)A}{\mu {{}^{\prime }}_v-\mu {{}^{\prime }}_r}and\omega \delta +{\omega }^{\prime }{\delta }^{\prime }=0$

Where $\omega$ and $\omega$' are the dispersive powers of the two prisms and d and d' their mean deviations.

(ix) Dispersion without deviation

A combination of two prisms in which deviation produced for the mean ray by the first prism is equal and opposite to that produced by the second prism is called a direct vision prism. This combination produces dispersion without deviation.

For deviation to be zero, ($\delta$ + $\delta$') = 0

$\Rightarrow \left(\mu -1\right)A+\left({\mu }^{\prime }-1\right)A^{\prime }=0$

$\Rightarrow A^{\prime }=-\frac{\left(\mu -1\right)A}{\left({\mu }^{\prime }-1\right)}\left(-ve\text{ sign }\Rightarrow \text{ prism A }{}^{\prime }\text{ has to be kept inverted}\right)$

Illustration 1: A prism of crown glass refracting angle of 50° and mean refractive index = 1.51 is combined with one flint glass prism of refractive index = 1.65 to produce no deviation. Find the angle of flint glass and net dispersion.

Given: $\mu$_v = 1.523, $\mu$_R = 1.513 (for crown glass)

$\mu$'_v = 1.665, $\mu$'_R = 1.645 (for flint glass)

Solution: Let A' be the angle of flint glass prism.

Here A = 5° and $\mu$ = 1.51 for crown glass prism.

$\delta =\left(\mu -1\right)A=\left(1.51-1\right)\times 5=2.55{}^{\circ }$

Deviation produced by flint glass

$\delta =\left({\mu }^{\prime }-1\right)A^{\prime }=\left(1.65-1\right)A^{\prime }=0.65A^{\prime }$

For no deviation $\delta$' = $\delta$

or 0.65 A' = 2.55

$A^{\prime }=\frac{2.55}{0.65}=3.92$

Net dispersion,

\begin{align}  \left( {{\mu }_{v}}-{{\mu }_{R}} \right)A-\left( \mu {{'}_{v}}-\mu {{'}_{R}} \right)A'=\left( 1.523-1.513 \right)5-\left( 1.655-1.645 \right)3.92 \\  =0.05-0.0784=-0.0284 \\ \end{align}