Question

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet rays are $\mu _{r},\mu _{y}$ and $\mu _{v}$ respectively and those for the flint glass are $\mu _{r}^{′},\mu _{y}^{′}$ and $\mu _{v}^{′}$ respectively. Find the ratio $\frac{A ′}{A}$ for which, there is no net angular dispersion ( $A$ and $A′$ are respective deviations)

• A. $\frac{\mu _{y} - 1}{\mu _{y}^{′} - 1}$
• B. $\frac{2 \left(\right. \left(\mu \right)_{v} - \left(\mu \right)_{r} \left.\right)}{\left(\right. \mu \left(′\right)_{v} - \mu \left(′\right)_{r} \left.\right)}$
• C. $\frac{\mu _{v} - \mu _{r}}{\mu ′_{v} - \mu ′_{r}}$
• D. $\frac{2 \left(\right. \left(\mu \right)_{y} - 1 \left.\right)}{\left(\right. \mu \left(′\right)_{y} - 1 \left.\right)}$

Answer 1

Answer: (B)

For no net angular dispersion
$\theta _{n e t}=\theta _{1}-\theta _{2}+\theta _{3}=0$
$\theta _{2}=\theta _{1}+\theta _{3}$
$\left(\mu _{v}^{′} - \mu _{r}^{′}\right)A^{′}=\left(\left(\mu \right)_{v} - \left(\mu \right)_{r}\right)A+\left(\left(\mu \right)_{v} - \left(\mu \right)_{r}\right)A$
$\frac{A ′}{A}=\frac{2 \left(\right. \left(\mu \right)_{v} - \left(\mu \right)_{r} \left.\right)}{\left(\right. \mu \left(′\right)_{v} - \mu \left(′\right)_{r} \left.\right)}$