Question

A sphere of radius $R$ made of material of refractive index ${\mu }_2$ is placed in a medium of refractive index ${\mu }_1$. Find the value of $x$, such that a real image is formed at a distance $x$ from the surface as shown in the figure below.

• A. $\left(\frac{{\mu }_2}{{\mu }_2-{\mu }_1}\right)R$
• B. $\left(\frac{{\mu }_1}{{\mu }_2-{\mu }_1}\right)R$
• C. $\left(\frac{{\mu }_2-{\mu }_1}{{\mu }_2}\right)R$
• D. $\left(\frac{{\mu }_2-{\mu }_1}{{\mu }_1}\right)R$

Answer 1

The correct option is B $\left(\frac{{\mu }_1}{{\mu }_2-{\mu }_1}\right)R$

If a real image is to be formed equidistant from the sphere, then the rays must pass symmetrical through the sphere, as shown in the figure.


For the refraction at first surface,

using the equation:

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$

If $x$ is the distance from the surface of the sphere at which point object should be placed then,

$\left(\frac{{\mu }_2}{\infty }\right)-$$\left(\frac{{\mu }_1}{-x}\right)$=$\left(\frac{{\mu }_2-{\mu }_1}{R}\right)$

$\therefore x=\left(\frac{{\mu }_1}{{\mu }_2-{\mu }_1}\right)R$

Thus option (b) is the correct answer.