Question

A prism of refractive index $n_1$ and another prism of refractive index $n_2$ are struck together without a gap as shown, $n_1$ and $n_2$ depend upon $\lambda$ the wave length of light according to$n_1=1.2+\frac{10.8\times {10}^4}{{\lambda }^2}$ and $n_2=1.45+\frac{1.8\times {10}^4}{{\lambda }^2}$, where $\lambda$ is in nanometer.
(a)Calculate the wave length ${\lambda }_0$ for which rays incident at any angle on the interface pass through without bending a the interface.
(b)For light of wave length ${\lambda }_0$ for which rays incident on the face $AC$ such that the deviated produced by the combination of the prism is minimum.

Answer 1

(a) Light rays travel undaviated (in a straight line)if the refractive index of the medium does not change for any value of angle of incidence
$\Rightarrow n_1=n_2$
$\Rightarrow 1.2+\frac{10.8\times {10}^4}{{\lambda }_0^2}=1.45+\frac{1.8\times {10}^4}{{\lambda }_0^2}$
$\Rightarrow {\lambda }_0=600mm$.
$n_0=1.2+\frac{10.8\times {10}^4}{{\lambda }_0^2}=(1.45+\frac{1.8\times {10}^4}{{\left(600\right)}^2})=1.5$
(b) If $n_1$ and $n_2$ are same the combination of prism will behave as a single prism of prism angle ${60}^o$
$\mu =\frac{\sin \frac{A+{\delta }_m}{2}}{\sin A/2}$
$\Rightarrow \sin \frac{A+{\delta }_m}{2}=1.5\times \frac{1}{2}=\frac{3}{4}$
$\Rightarrow \frac{A+{\delta }_m}{2}={48.6}^o\Rightarrow \delta m=37.2$
$r={30}^o,sini=1.5\sin {30}^o=0.75$
$\Rightarrow l={\sin }^{-1}0.75={48.6}^o$