Question

A small object stuck on the surface of a glass sphere $(n=1.5)$ is viewed from the diametrically opposite position. Find transverse magnification if the surrounding medium is air.

• A. $3$
• B. $-2.5$
• C. $2$
• D. $2.5$

Answer 1

The correct option is A $3$

Given that,
Refractive index of the glass, $n_1=n=1.5$
Refractive index of air, $n_2=1$
Let the radius of the sphere is $R$ and image of the object is formed at $v$.


Sign convention: Direction along the rays is taken as positive.
Since the object is viewed from diametrically opposite end, $u=-2R$

For the refraction at spherical interface,
$\frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R}$
Here $R=-R$ as per sign convention

$\Rightarrow \frac{1}{v}-\frac{1.5}{(-2R)}=\frac{1-1.5}{-R}$
$\Rightarrow \frac{1}{v}+\frac{1.5}{(-2R)}=\frac{0.5}{R}$
$\Rightarrow \frac{1}{v}=\frac{0.5}{\left(R\right)}-\frac{1.5}{2R}=\frac{1-1.5}{2R}$
$\therefore v=-4R$
The transverse magnification is given by
$m=\frac{n_{1}v}{n_{2}u}$
$\Rightarrow m=\frac{\left(1.5\right)(-4R)}{\left(1\right)(-2R)}$
$\therefore m=3$
Hence, option (a) is correct.