The dispersive powers of crown and flint glasses are $0.03$ and $0.05$ respectively. The refractive indices for yellow light for these glasses are $1.517$ and $1.621$ respectively. It is desired to form an achromatic combination of prisms of crown and flint glasses which can produce a deviation of $1^{\circ }$ in the yellow ray. Find the refracting angles of the two prisms needed.

The correct option is D ${4.8}^{\circ },{2.4}^{\circ }$
Given,
$\omega =0.03;{\omega }^{\prime }=0.05$
${\mu }_y=1.517;{\mu }_y^{\prime }=1.621$

Where, $\omega ,{\omega }^{\prime }=\text{Dispersive powers of crown and flint prisms respectively}$
${\mu }_y,{\mu }_y^{\prime }=\text{R.I. of the yellow light for crown and flint prisms respectively}$

Let, the angle of the crown and flint prism needed is $A$ and $A^{\prime }$ respectively.

Dispersive power is,

$\omega =\frac{{\mu }_v-{\mu }_r}{{\mu }_y-1}$

${\mu }_v-{\mu }_r=({\mu }_y-1)\omega$

The angular dispersion produced by the crown prism is,

$({\mu }_v-{\mu }_r)A=({\mu }_y-1)\omega A$

Similarly, the angular dispersion produced by the flint prism is,

$({\mu }_v^{\prime }-{\mu }_r^{\prime })A^{\prime }=({\mu }_y^{\prime }-1){\omega }^{\prime }{A}^{{}^{\prime }}$

For achromatic combination, the net dispersion should be zero. Thus,

$({\mu }_y-1)\omega A=({\mu }_y^{\prime }-1){\omega }^{\prime }{A}^{\prime }$

$\frac{A^{{}^{\prime }}}{A}=\frac{({\mu }_y-1)\omega }{({\mu }_y^{\prime }-1){\omega }^{\prime }}=\frac{0.517\times 0.03}{0.621\times 0.05}$

$\frac{A^{{}^{\prime }}}{A}=0.5.......\left(1\right)$

The deviation in the yellow ray produced by the crown prism is,

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

And by the flint prism is,

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

The net deviation produced by the combination is,

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

$1^{\circ }=0.517A-0.621A^{\prime }......\left(2\right)$

Solving $\left(1\right)$ and $\left(2\right)$ we get,

$A={4.8}^{\circ }$ and $A^{\prime }={2.4}^{\circ }$

Thus, the crown prism should have its refracting angle ${4.8}^{\circ }$ and that of the flint prism should be ${2.4}^{\circ }$.

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

Why this question ? Note: Achromatic combination means that colours of the white light are not separated by the combination of prisms. Therefore, achromatic combination refers to the deviation without dispersion.