Question

A ray of light incident on a transparent sphere at an angle $\pi /4$ and refracted at an angle r, emerges from the sphere after suffering one internal reflection. The total angle of deviation of the ray is$?$

• A. $\frac{3\pi }{2}-4r$
• B. $\frac{\pi }{2}-4r$
• C. $\frac{\pi }{4}-r$
• D. $\frac{5\pi }{2}-4r$

Answer 1

Answer: (D)

$\frac{5\pi }{2}-4r$

Observing the above figure, it is given that $\theta =\pi /4$.

From $△OAB$, as $OA=OB$, we have $r_1=r_2=r$.

Similarly, from laws of reflection $\angle OBC=r$ and thus, $\angle BCO=r$

The deviation of light ray in first refraction is $D_1=\theta -r_1=\frac{\pi }{4}-r$

The deviation of the ray in Total internal reflection is $D_2=2\pi -2r_2=2\pi -2r$

The deviation of the ray in the second refraction is $D_3=\frac{\pi }{4}-r$

Thus, the total deviation of the ray is $D_1+D_2+D_3=2(\frac{\pi }{4}-r)+2\pi -2r=\frac{5\pi }{2}-4r$