Question

A right angles prism $({45}^o,{90}^o,{45}^o)$ of refractive index n has a plate of refractive index $(n_1<n)$ cemented to its diagonal face. The assembly is in air. A ray is incident on AB.
If the angle of incidence at AB for which the ray strikes the diagonal face at the critical angle is X in degrees. Find X?

Answer 1

$sinC=\frac{n_1}{n}$
From fig, $({90}^o-r_1)+{45}^o+({90}^o-C)={180}^o$
$\Rightarrow r_1={45}^o-C$
From Snell's law,
$\frac{sini}{sin{r}_1}=n$
$\therefore sini=nsin{r}_1$
$=nsin({45}^o-C)$
$=n(sin{45}^{o}cosC-cos{45}^{o}sinC)$
$=\frac{n}{\sqrt{2}}(cosC-sinC)$
$=\frac{n}{\sqrt{2}}[\sqrt{1-si{n}^{2}C}-sinC]$
$=[\sqrt{1-{\left(\frac{n_1}{n}\right)}^2}-\frac{n_1}{n}]$
$\frac{1}{\sqrt{2}}[\sqrt{(}{n}^2-n_1^2)-n_1]$
$\Rightarrow i=si{n}^{-1}\left\{\frac{1}{\sqrt{2}}\right({\sqrt{n}}^2-n_1^2)-n_1\}$