Question

The plane surface of a plano- convex lens is silvered. If $\mu$ be the refractive index and $R$, the radius of curvature of curved surface, then the system will behave like a concave mirror of radius of curvature of

• A. $\mu R$
• B. $\frac{R}{(\mu -1)}$
• C. $\frac{R^2}{\mu }$
• D. $\left(\frac{\mu +1}{\mu -1}\right)R$

Answer 1

The correct option is B $\frac{R}{(\mu -1)}$

Let $f_l$ and $f_m$ be the focal lengths of convex lens and the plane mirror (plane polish surface) respectively, then combined power of system,
$P=P_M-2P_L$

$\frac{1}{F}=\frac{1}{f_m}-\frac{2}{f_l}$

Since, $f_p=\infty$, So

$\Rightarrow \frac{1}{F}=-\frac{2}{f_l}....\left(1\right)$

Now, from lens maker's formula for plano-convex lens,

$\frac{1}{f_l}=\frac{(\mu -1)}{R}.....\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$,

$\Rightarrow \frac{1}{F}=-\frac{2(\mu -1)}{R}$

$\Rightarrow F=-\frac{R}{2(\mu -1)}$

Here negative sign of effective focal length indicates that combination acts as concave mirror.

Therefore,

the radius of curvature $=2|F|$

$=2\left(\frac{R}{2(\mu -1)}\right)$

$=\frac{R}{(\mu -1)}$