Question

Consider a sphere of radius R made of glass of refractive index $\mu$. A small object moves along the diameter with a constant velocity u. Find the velocity of the image as seen by an observer outside when the object passes through center

• A. $u$
• B. $\mu u$
• C. $u/\mu$
• D. $0$

Answer 1

The correct option is B $\mu u$

Here x is the position of object from the refracting surface, x' is the position of its image after refraction(which is seen by the observer).

Knowing refraction from curved surface,

$\frac{1}{x^{\prime }}-\frac{\mu }{x}=\frac{1-\mu }{-R}$

$x^{\prime }=\frac{1}{\frac{\mu }{x}+\frac{\mu -1}{R}}$

Velocity of image=$\frac{d{x}^{\prime }}{dt}$=$\frac{-1}{(\frac{\mu }{x}+\frac{\mu -1}{R}{)}^2}\frac{-\mu }{x^2}\frac{dx}{dt}$

Velocity when object passes through center $={\frac{d{x}^{\prime }}{dt}}_{x=-R}=\mu \frac{dx}{dt}=\mu u$