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 $1.5P$ is

• A. $R/3$
• B. $3R/2$
• C. $R/2$
• D. $2R$

Answer 1

The correct option is A

$R/3$

Step 1: Given Data And Assumptions:

Power of double convex lens $=P$

Radius of curvature of double convex lens $=R$

Power of plano-convex lens $=1.5P$

Let, Radius of plano-convex lens is $R\text{'}$ and $\infty$

Step 2: Formula used:

$P=\frac{1}{f}$

Where, $p=$power of lens

$f=$ focal length of lens

Lens maker's formula:

$\frac{1}{f}=\left(n-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Where, $f=$ Focal length of lens

$n=$Refractive index of material used

$R_1\&R_2=$Radius of curvature of lens

Step 3: Calculating the radius of curvature of plano-convex lens:

For double convex lens:

$\begin{array}{rcl}P& =& \left(n-1\right)\left(\frac{1}{R}-\left(-\frac{1}{R}\right)\right)\\ & =& \left(n-1\right)\left(\frac{2}{R}\right)..................\left(1\right)\end{array}$

For plano-convex lens:

$\begin{array}{rcl}1.5P& =& \left(n-1\right)\left(\frac{1}{R\text{'}}-\frac{1}{\infty }\right)\\ & =& \left(n-1\right)\frac{1}{R\text{'}}..................\left(2\right)\end{array}$

Equation $\left(2\right)/\left(1\right)$

$\begin{array}{rcl}1.5& =& \left(\frac{1}{R\text{'}}\right)/\left(\frac{2}{R}\right)\\ 1.5& =& \frac{R}{2R\text{'}}\\ R\text{'}& =& \frac{R}{2\times 1.5}\\ & =& \frac{R}{3}\\ & & \end{array}$

Hence, option A is the correct answer.