Question

Focal length of the plano-convex lens is _____ when its radius of curvature of the surface is $R$ and $n$ is the refractive index of the lens
A) $f = R$
B)$f = \dfrac{R}{2}$
C) $f = \dfrac{R}{{n - 1}}$
D) $f = \dfrac{{n - 1}}{R}$

Answer 1

Plano-convex lenses have one surface that is straight while the other surface is convex. The lens maker formula is useful in determining the focal length of different kinds of lenses.

Formula used: $\dfrac{1}{f}\, = \left( {\dfrac{{{\mu _2}}}{{{\mu _1}}}\, - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$ where ${\mu _1}$ and ${\mu _2}$ are the refractive index of the surrounding medium and the lens, respectively and ${R_1}$ and ${R_2}$ are the radius of curvature of the two surfaces of the lens.

Complete step by step solution:
A plano-convex lens has one plane surface and one flat surface. Hence, we can define the radius of the curvature for the flat surface as ${R_1} = \infty$ and for the curved surface as $\Rightarrow {R_2} = - R$.
Since we haven’t been given any information about the surrounding medium, we can assume that the lens is surrounded by air which has a refractive index ${\mu _1} = 1$ and the lens has a refractive index given as ${\mu _2} = n$.
From the lens maker formula, we have:
$\Rightarrow \dfrac{1}{f}\, = \left( {\dfrac{{{\mu _2}}}{{{\mu _1}}}\, - 1} \right)\left( {\dfrac{1}{{{R_1}}} - \dfrac{1}{{{R_2}}}} \right)$
Substituting ${\mu _1} = 1$, ${\mu _2} = n$, ${R_1} = \infty$ and ${R_2} = - R$, we get,
$\Rightarrow \dfrac{1}{f}\, = \left( {n\, - 1} \right)\left( {\dfrac{1}{\infty } - \dfrac{1}{{ - R}}} \right)$
Since $\dfrac{1}{\infty } = 0$, we can write,
$\Rightarrow \dfrac{1}{f}\, = \left( {n\, - 1} \right)\left( {\dfrac{1}{R}} \right)$
Taking the reciprocal on both sides, we get the focal length as:
$\therefore f = \dfrac{R}{{n - 1}}$ which corresponds to option C.
Hence, the correct option is option (C).

Note:
While using the lens maker formula, we must be careful in using sign conventions of the radius of curvature. The plane surface has infinite radius of curvature since it doesn’t curve while the convex surface has a negative radius of curvature. We must always assume the surrounding medium as air unless specifically mentioned in the question. The lens maker formula used here assumes that the lens is very thin compared to the radius of curvature of the convex surface.