Question

Consider a transparent hemisphere $(n=2)$ in front of which a small object is placed in air $(n=1)$ as shown.
Consider a ray starting from $O$ which strikes the spherical surface at grazing incidence $(i={90}^{\circ })$. Taking $x=R$, what will be the angle (from normal) at which the ray may emerge from the plane surface.

• A. ${90}^{\circ }$
• B. $0^{\circ }$
• C. ${30}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

Answer: (A)

${90}^{\circ }$

Using Snell's law at point A, $\sin {90}^{\circ }=2\sin \angle CAB$

$\angle CAB={30}^{\circ }$

Also, $\angle ACO={60}^{\circ }$(from geometry)

Hence,$\angle CA{B}^{\prime }={60}^{\circ }$

$\angle B^{\prime }AB={60}^{\circ }-{30}^{\circ }={30}^{\circ }$

hence angle of incidence at plane surface= ${30}^{\circ }$(alternate interior angles)

Using snell's law at plane surface,

$2\sin {30}^{\circ }=\sin e$

Hence $e={90}^{\circ }$