Question

If $f_1$ and $f_2$ represent the first and second focal lengths of a single spherical refracting surface of refractive index μ, then
(A) ${f_1}{f_2} = \mu $
(B) $\dfrac{{{f_1}}}{{{f_2}}} = - 1$
(C) ${f_1} = 2{f_2}$
(D) ${f_2} = \mu {f_1}$

Answer 1

Hint
The angle of refraction, and that implies the focal length, depends on the geometry of the lens surface as well as the material used to construct the lens. Materials with a high index of refraction will have a shorter focal length compared to that of a lens with a lower refractive index. We need to understand the relation between the refractive index and the focal length of the spherical surface.
$\Rightarrow \dfrac{1}{{{f_1}}} = \dfrac{{\left( {\mu - 1} \right)}}{R}$
$\Rightarrow \dfrac{1}{{{f_2}}} = \dfrac{{\left( {\mu - 1} \right)}}{{\mu R}}$
Where, the refractive index of the spherical surface is μ and the focal lengths are $f_1$ and $f_2$.

Complete step by step answer
As mentioned above, let us consider the refractive index of the spherical surface be μ and the focal lengths be $f_1$ and $f_2$.
As it is a single surface, it has only one radius of curvature.
Thus, the relation between the focal length, radius of curvature and the refractive index comes as,
$\Rightarrow \dfrac{1}{{{f_1}}} = \dfrac{{\left( {\mu - 1} \right)}}{R}$ …..eq.1
And,
$\Rightarrow \dfrac{1}{{{f_2}}} = \dfrac{{\left( {\mu - 1} \right)}}{{\mu R}}$ …..eq.2
Now, when we divide eq.1 by eq.2,
$\Rightarrow \dfrac{{\dfrac{1}{{{f_1}}}}}{{\dfrac{1}{{{f_2}}}}} = \dfrac{{\dfrac{{\left( {\mu - 1} \right)}}{R}}}{{\dfrac{{\left( {\mu - 1} \right)}}{{\mu R}}}}$
$\Rightarrow \dfrac{{{f_2}}}{{{f_1}}} = \dfrac{{\left( {\mu - 1} \right)}}{R} \times \dfrac{{\mu R}}{{\left( {\mu - 1} \right)}}$
Thus, the relation becomes,
$\Rightarrow \dfrac{{{f_2}}}{{{f_1}}} = \mu $
$\Rightarrow {f_2} = \mu {f_1}$
Hence, the correct answer is option (D).

Note
There are many ways to find out the value of focal lengths of a spherical surface. We need to first find out the radius of curvature of the surface and divide by 2, to get the focal length. We can find the ratio of the focal lengths by using the refractive index also, as seen above.