Question

Two identical thin plano-convex lenses of refractive index $\mu$ are silvered, one on the plane side and the other on the convex side. The ratio of their focal lengths is

• A. $\mu$
• B. $\frac{\mu +1}{\mu }$
• C. $\frac{\mu }{\mu -1}$
• D. $\frac{\mu }{\mu +1}$

Answer 1

The correct option is C $\frac{\mu }{\mu -1}$

Case $\left(i\right)-$ silvered on the plane side. $\frac{1}{F_1}=\frac{2}{f}-\frac{1}{f_m}$
For plane mirror $f_m=\infty$
$\frac{1}{F_1}=\frac{2}{f}=2\frac{(\mu -1)}{R}$
$F_1=\frac{R}{2(\mu -1)}\to \left(1\right)$

Case $\left(ii\right)-$ Silvered on the convex side.
$\frac{1}{F_2}=\frac{2}{f}-\frac{1}{f_m}$
where $f_m$ is the tocal length of concave mirror.

$f_m=-\frac{R}{2}$
$\frac{1}{F_2}=\frac{2(\mu -1)}{R}+\frac{2}{R}=\frac{2\mu }{R}$
$F_2=\frac{R}{2\mu }\to \left(2\right)$

Rotio of focal length is
$\frac{F_1}{F_2}=\frac{R}{2(\mu -1)}\times \frac{2\mu }{R}$
$\frac{F_1}{F_2}=\frac{\mu }{\mu -1}$