Question

Three thin prisms are combined as shown in figure. The refractive indices of the crown glass for red, yellow and violet lights are ${\mu }_r,{\mu }_y$ and ${\mu }_v$ respectively and those for the flint glass are ${\mu }_r^{{}^{\prime }},{\mu }_y^{{}^{\prime }}$ and ${\mu }_v^{{}^{\prime }}$ respectively. Find the $\frac{A^{{}^{\prime }}}{A}$ for which, there is no net angular dispersion.

• A. $\frac{{\mu }_v-{\mu }_r}{{\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }}}$
• B. $2\left(\frac{{\mu }_v-{\mu }_r}{{\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }}}\right)$
• C. $\frac{{\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }}}{{\mu }_v-{\mu }_r}$
• D. $2\left(\frac{{\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }}}{{\mu }_v-{\mu }_r}\right)$

Answer 1

The correct option is B $2\left(\frac{{\mu }_v-{\mu }_r}{{\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }}}\right)$

Total deviation suffered by the violet light by these prisms combination is,

${\delta }_v={\delta }_{cv}-{\delta }_{fv}+{\delta }_{cv}$

Where, ${\delta }_{cv}=\text{Deviation suffered by the violet light in the crown glass prism}{\delta }_{fv}=\text{Deviation suffered by the violet light in the flint glass prism}$

$\Rightarrow {\delta }_v=2{\delta }_{cv}-{\delta }_{fv}$

As we know, the minimum deviation suffered is,

${\delta }_m=(\mu -1)A$

$\Rightarrow {\delta }_v=2A({\mu }_v-1)-A^{{}^{\prime }}({\mu }_v^{{}^{\prime }}-1)$

Similarly, the deviation suffered by the red light is,

${\delta }_r=2A({\mu }_r-1)-A^{{}^{\prime }}({\mu }_r^{{}^{\prime }}-1)$

For net angular dispersion to be zero,

${\delta }_v-{\delta }_r=0$

$\Rightarrow 2A({\mu }_v-1)-A^{{}^{\prime }}({\mu }_v^{{}^{\prime }}-1)-\left[2A\right({\mu }_r-1)-A^{{}^{\prime }}({\mu }_r^{{}^{\prime }}-1\left)\right]=0$

$\Rightarrow 2A({\mu }_v-{\mu }_r)=A^{{}^{\prime }}({\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }})$

$\Rightarrow \frac{A^{{}^{\prime }}}{A}=\frac{2({\mu }_v-{\mu }_r)}{({\mu }_v^{{}^{\prime }}-{\mu }_r^{{}^{\prime }})}$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(B\right)$ is the correct answer.

Why this question ? Note: If the second prism in the question is also made up of the crown glass instead of flint glass and if $(A=A^{{}^{\prime }})$, then the combination of three prisms will behave as a single prism.