Question

A ray of light strikes a plane mirror at an angle of incidence ${45}^o$ as shown in fig. 1.364. After reflection, the ray passes through a prism of refractive index 1.5 whose apex angle is $4^o$. If the mirror be rotated by X degrees then the total deviation of the ray becomes ${90}^o$. Find X?

Answer 1

The deviation produced by small angled prism,
${\delta }_1=(\mu -1)\alpha =(1.5-1)4^o=2^o$ (always)
Deviation caused by mirror,
${\delta }_2={180}^o-2i={180}^o-2\times {45}^o={90}^o$
Net deviation produced by system $={\delta }_1+{\delta }_2=2^o+{90}^o={92}^o$
This is more than ${90}^o$.
Greater is angle of incidence on the mirror, smaller is the deviation.
If $\beta$ is the angle of rotation of mirror in clockwise direction to increase angle of incidence, then deviation produced by the mirror will be ${180}^o-2({45}^o+\beta )={90}^o-2\beta$
Therefore, total deviation produced $={90}^o-2\beta +2^o={92}^o-2\beta$
Given, ${92}^o-2\beta ={90}^o\Rightarrow \beta =1^o$