Question

A portion of straight glass rod of diameter $4 \,cm$ and refractive index $\mu=2$ is bent into an arc of circle and a parallel beam of light is incident on it as shown in the figure. Find the smallest value of $R$, in $cm$ (the radius of outer arc) which permits all the light to pass around the arc.

Answer 1

The incident angle on the bent surface is the smallest for the ray shown in figure. If this angle is greater than critical angle then all the rays including this ray will be totally reflected.

$\therefore \theta \geq C \Rightarrow \sin \theta \geq \sin C$
$\frac{ R -4}{ R } \geq \frac{1}{\mu}$
$\mu( R -4) \geq R$
$\mu R -4 \mu \geq R $
$\Rightarrow \mu R - R \geq 4 \mu$
$R \geq \frac{4 \mu}{\mu-1}$
$\therefore R _{\min }=\frac{4 \mu}{\mu-1}$
$=\frac{4 \times 1 \cdot 5}{1 \cdot 5-1}=12\, cm$