Question

A uniform, horizontal beam of light is incident upon a prism as shown in the fig. The prism is in the shape of a quarter cylinder of radius R $=$ 5cm, and has a refractive index $\frac{2}{\sqrt{3}}$. A patch on the table for a distance x from the cylinder is unilluminated. The value of x is

• A. $2.5$ cm
• B. $5$ cm
• C. $5\sqrt{3}$cm
• D. $10$ cm

Answer 1

The correct option is B $5$ cm

Figure-(I) shows how for rays nearer to base are refracted but as rays move away from base, they fall nearer and after a certain distance above base, all rays are internally reflected.

So, we have to find the distance x where there is no light. As on increasing height rays fall nearer to the prism, so we have to find where will the ray for critical angle of incidence will intersect base line. And inside that point there will be no illumination as for larger angle of incidence , rays are internally reflected.

From figure-2, it is clear that $i=C$=critical angle and refracted ray for this goes perpendicular to normal which is radius $OA$ here. Refracted ray is $AB$.

$i=C={\sin }^{{}_1}\frac{1}{\mu }={\sin }^{-1}\frac{\sqrt{3}}{2}={60}^o$

From $\Delta OAB:-$

$\cos i=\frac{OA}{OB}=\frac{R}{R+x}$

$⟹\cos {60}^o=\frac{1}{2}=\frac{5}{5+x}$

$⟹x=5cm$

Answer-(B)