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  4. A transparent sphere of radius R and refractive index μ is kept in air. At what distance from the surface of the sphere should a point object be placed so as to form real image at the same distance from the sphere?

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Q. A transparent sphere of radius R and refractive index $\mu$ is kept in air. At what distance from the surface of the sphere should a point object be placed so as to form real image at the same distance from the sphere?

Ray Optics and Optical Instruments

Solution:

As $P_{1} O=P_{2} I$, therefore, the rays must pass through the sphere in a direction parallel to the optic axis, as shown in the figure.
image
Now, for first refraction at $A$ from rarer to denser medium,
$-\frac{\mu_{1}}{u}+\frac{\mu_{2}}{v}=\frac{\mu_{2}-\mu_{1}}{R}$
$ \Rightarrow -\frac{1}{u}+\frac{\mu}{\infty}=\frac{\mu-1}{R} $
$\therefore u=-\frac{R}{(\mu-1)}$
or $\quad|u|=\frac{R}{(\mu-1)}$