A double convex lens has power P and same radii of curvature R of both the surfaces.

Question:

A double convex lens has power $P$ and same radii of curvature $R$ of both the surfaces. The radius of curvature of a surface of a plano-convex lens made of the same material with power $1.5 P$ is :

  1. $2 R$

  2. $\frac{R}{2}$

  3. $\frac{3 R}{2}$

  4. $\frac{R}{3}$


Correct Option: 4

Solution:

(4) Given, using lens maker's formula

$\frac{1}{f}=(k-1)\left(\frac{1}{R_{1}}-\frac{1}{R_{2}}\right)$

Here, $R_{1}=R_{2}=R$ (For double convex lens)

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

$\Rightarrow P=\frac{1}{f}=(\mu-1) \frac{2}{R}$             .....(i)

For plano convex lens,

$R_{1}=R^{\prime}, R_{2}=\infty$

Using lens maker's formula again, we have

$1.5 P=(\mu-1)\left(\frac{1}{R^{\prime}}-\frac{1}{\infty}\right)$                 ....(ii)

$\Rightarrow \frac{3}{2} P=\frac{\mu-1}{R^{\prime}}$

From (i) and (ii),

$\frac{3}{2}=\frac{R^{\prime}}{2 R} \Rightarrow R^{\prime}=\frac{R}{3}$

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