Question

Two prisms of prism angles A and A' are placed in contact with each other with second parts upside down. the refractive indices for violet and red colors for their materials are $\mu =1.7$,$\mu =1.751$ and $\mu =1.758$, $\mu =1.754$. a white light ray when made to fall upon combination from one side emerges from the other side such that the constituents color rays are parallel t each other. If A=$5^{\circ }$, then A'=

• A. $5^{\circ }$
• B. ${10}^{\circ }$
• C. ${20}^{\circ }$
• D. ${2.5}^{\circ }$

Answer 1

The correct option is A $5^{\circ }$

$\begin{array}{c}\text{We know,}\\ \delta =(\mu -1)A-\left(i\right)\\ \delta \to \text{angle of deviation}\\ \mu \to \text{refractive index of prism}\\ A\to \text{angle of prism in radian}\end{array}$

$\begin{array}{c}\text{since light rays are parallel,}\\ \Rightarrow \text{The net deviation through}\\ \text{The prisms is same for}\\ \text{violet and red light.}\end{array}$

$\begin{array}{c}\text{For red light,}\\ \delta =({\mu }_1-1)A_1+({\mu }_2-1)A_2\end{array}$

$\Rightarrow \delta =(1.758-1)\frac{5\pi }{180}+(1.754-1)A^{\prime }-\left(ii\right)$

$\begin{array}{c}\text{For violet light,}\end{array}$

$\Rightarrow \delta =(1.7-1)\frac{5\pi }{180}+(1.75-1)A^{\prime }-\left(iii\right)$

$\begin{array}{c}\text{from (ii) and (iii)}\\ \begin{array}{c}\left(0.758\right)\frac{5x}{180}+0.754A^{\prime }=0.7\times \frac{5\pi }{180}+0.751A^{{}^{\prime }}\end{array}\end{array}$

$\Rightarrow 0.058\times \frac{5\pi }{180}=0.054A^{{}^{\prime }}$

$\begin{array}{c}\Rightarrow A^{\prime }=\frac{0.051}{0.054}\times 5\times \frac{\pi }{180}\text{rad}\\ \Rightarrow A^{\prime }=5^{\circ }\end{array}$