Question

The refractive indices of the crown glass for blue and red lights are 1.51 and 1.49, respectively, and those of the flint glass are 1.77 and 1.73, respectively. An isosceles prism of angle $6^o$ is made of crown glass. A beam of white light is incident at a small angle on this prism. The other flint glass isosceles prism is combined with the crown glass prism such that there is no deviation of the incident light. If the net dispersion of the combined system is $\frac{x}{100}$. Find $x$.

Answer 1

Deviation produced by a prism $=({\mu }_y-1)\alpha$.
For no deviation caused by combination of two prisms,
$({\mu }_y-1)\alpha +({\mu }_y^{\prime }-1){\alpha }^{\prime }=0$
$\Rightarrow \alpha =-\frac{{\mu }_y-1}{{\mu }_y^{\prime }-1}\alpha$
Here $\alpha ={60}^o$
${\mu }_y^{\prime }=\frac{{\mu }_b^{\prime }+{\mu }_r^{\prime }}{2}=\frac{1.77+1.73}{2}=1.75$
Refracting angle of flint glass prism,
${\alpha }^{\prime }=\frac{{\mu }_y^{\prime }-1}{{\mu }_y^{\prime }-1}\alpha =-\left[\frac{(1.50-1)}{(1.75-1)}\right]\times 6^o$
$=-\frac{0.50}{0.75}\times 6^o=-4^o$
The negative sign shows that the bases of prisms are opposite.
Therefore, refracting angle of flint glass prism $=4^o$
Net dispersion produced
$\Delta \theta =({\delta }_b-{\delta }_r)+({\delta }_b^{\prime }-{\delta }_r^{\prime })$
$=({\mu }_b-{\mu }_r)\alpha +({\mu }_b^{\prime }-{\mu }_r^{\prime }){\alpha }^{\prime }$
$=(1.52-1.49)\times 6^o-(1.77-1.73)\times 4^o$
$={0.12}^o-{0.16}^o=-{0.04}^o={0.04}^o$ (numerically)