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 ${\mu }_r^{{}^{\prime }},{\mu }_y^{{}^{\prime }}$ and ${\mu }_v^{{}^{\prime }}$ respectively. Find $\frac{A^{{}^{\prime }}}{A}$ for which, there is no net deviation in the yellow light.

The correct option is B $\frac{2({\mu }_y-1)}{{\mu }_y^{{}^{\prime }}-1}$
The total deviation for the yellow ray produced by the prism combination is,

${\delta }_y={\delta }_{cy}-{\delta }_{fy}+{\delta }_{cy}=2{\delta }_{cy}-{\delta }_{\delta y}$

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

As we know, the minimum deviation suffered is,

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

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

For net deviation in the yellow ray to be zero,

${\delta }_y=0$

$\Rightarrow 2({\mu }_y-1)A-({\mu }_y^{\prime }-1)A^{\prime }=0$

$\therefore \frac{A^{\prime }}{A}=\frac{2({\mu }_y-1)}{{\mu }_y^{\prime }-1}$

<!--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.