Question

A ray of light is falling on a glass sphere of $\mu =\sqrt{3}$ such that the incident ray and the emergent ray, when produced, intersect at a point on the surface of the sphere. Find the value of angle of incidence in degrees.

Answer 1

As shown in the figure, refraction occurs at points A and B on the glass sphere.

Both the angles of refraction have to be same, since they are angles opposite to the equal sides of the triangle ABC.

This gives $\angle DAB=i-r$ and $\angle DBA=i-r$

In triangle ABC, $\angle ACB=\pi -2r$

Hence, $\angle AEB=\frac{\pi }{2}-r$

In triangle ABD, $\angle ADB=\pi -2(i-r)$

Sum of opposite angles of a cyclic quadrilateral is $\pi$.

$\angle ABD+\angle AEB$

$2i-r=\frac{\pi }{2}$

Using snell's law at point A gives, $\sin i=\mu \sin r$

Hence $i={60}^{\circ }$