Question

The radius of curvature of a plano-convex lens is 20cm. If the refractive index of material of the lens is 1.5 it will
A.) Act as a convex lens only for the objects that lie on its curved side.
B.) Act as a concave lens for the object that lies on its curved side.
C.) Act as a convex lens irrespective of the side on which the object lies.
D.) Act as a convex lens irrespective of the side on which the object lies.

Answer 1

Hint: Using the concept when the lens is in a medium and has a different radius of curvature. Considering the objects can be placed on both sides of the lens, we find the focal length from each side which tells us about what we will observe.

Formula used:\[\dfrac{1}{f}=(\mu -1)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)\]

Complete step by step solution:
In a plano-convex lens the radius of curvature of the plane side being ${{R}_{1}}=\infty $ is equal to infinity. As, the mirror needs to be part of the sphere and for it to be flat the radius of the sphere has to be infinity.

Radius of curvature of the convex side is ${{R}_{2}}=-20cm$ as given in the question.

Now, lets first consider the object is on the plane side of the lens using the formula

\[\dfrac{1}{f}=(\mu -1)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right)\]
$\dfrac{1}{f}=\left( 1.5-1 \right)\left( \dfrac{1}{\infty }-\dfrac{1}{\left( -20 \right)} \right)$

$\begin{align}
  & \dfrac{1}{f}=(0.5)(0.05) \\
 & f=40cm \\
 & \\
\end{align}$

We used ${{R}_{2}}=-20cm$ because the vertex lies on the right side of the centre of curvature.

Now, taking the case when the object lies on the curved side we will still use the same formula but in this case${{R}_{1}}=20cm$. This time it is positive because the vertex is on the left side of the centre of curvature.${{R}_{2}}$ will be equal to infinity.

$\begin{align}
  & \dfrac{1}{f}=\left( \mu -1 \right)\left( \dfrac{1}{{{R}_{1}}}-\dfrac{1}{{{R}_{2}}} \right) \\
 & \dfrac{1}{f}=\left( 1.5-1 \right)\left( \dfrac{1}{20}-\dfrac{1}{\infty } \right) \\
 & \dfrac{1}{f}=\left( 0.5 \right)\left( 0.05 \right) \\
 & f=40cm \\
\end{align}$

In both the cases we are getting the same focal length that is equal to 40cm so that shows that the plano-convex lens will act as a convex lens irrespective of the side of the object. The lens is convex because the focal length is positive.

Hence, The correct answer to this question is option (c).

Note: Students should learn a little about the sign convention of the radius of curvature for different lenses and also about focal length. Don’t miss the radius of curvature of the plane side of the lens that is infinity, otherwise we will get the wrong solution that is a concave lens instead of the convex lens.