Question

The convex surface of a plano-convex lens is silvered and the radius of curvature of this surface is $R$. Find the focal length of the system.

Answer 1

$P_{finalmirror}=P_{lens}+P_{concavemirror}+P_{lens}$
As the ray of light first goes through lens then mirror then again lens.
$P_{finalmirror}=\frac{1}{f_{\ell }}-\frac{1}{f_m}+\frac{1}{f_{\ell }}=\frac{2}{f_{\ell }}-\frac{1}{f_m}$ where $f_{\ell }$ is given as
$\frac{1}{f_{\ell }}=(\mu -1)(\frac{1}{\infty }-\frac{1}{-R})=+\frac{(\mu -1)}{R}$
$\Rightarrow f_{\ell }=\frac{+R}{\mu -1}=2R$ for $\mu =1.5$
$f_m=-\frac{R}{2};P_{finalmirror}=\frac{2}{2R}+\frac{2}{R}=\frac{3}{R}$
$F_{finalmirror}=-\frac{1}{P_{finalmirror}}=-\frac{R}{3}$
and hence the final mirror will be concave.