Question

A lens forms a sharp image on a screen. On inserting a parallel-sided slab of glass between the lens and the screen it is found necessary to move the screen a distance $d$ away from the lens for the image to be again sharply focused. If the refractive index of glass relative to air is $\mu$ , then the thickness of the slab is
(A) $\dfrac{d}{\mu }$
(B) $\dfrac{\mu }{d}$
(C) $\dfrac{\mu d}{\mu -1}$
(D) $\dfrac{(\mu -1)d}{\mu }$

Answer 1

We have been asked to find the thickness of the slab in this question. To know that we need to know about the situation before and after the slab insertion. Before slab insertion, the image was being formed at a certain distance from the lens. Now due to the slab being inserted, the image shifts through a certain given distance. To focus the image again, the screen had to be moved by a distance equal to the distance of the shifting of the image.

Complete step by step answer:
We have been given the refractive index and the distance through which the screen had to be shifted.
The refractive index of the lens is $\mu$ and the screen was shifted by a distance $d$.
Let the thickness of the slab be $t$. The normal shift caused by the slab is equal to the distance through which the screen was shifted, that is, $d$
We can relate the normal shift with the thickness of the slab and the refractive index of the lens as
$d=t\left( 1-\dfrac{1}{\mu } \right)$ where the meanings of the symbols have been given above.
Rearranging the terms, we get

& d=t\dfrac{(\mu -1)}{\mu } \\\ & \Rightarrow \mu d=t(\mu -1) \\\ & \Rightarrow t=\dfrac{\mu d}{(\mu -1)} \\\ \end{aligned}$$ Hence the correct option is (C). **Note:** We have two types of shifts, namely lateral shift and normal shift. Students often confuse these two and hence it is important to know the difference between the two. The lateral shift can be defined as the perpendicular distance of the incident ray and the emergent rays, whereas normal shift may be described as the apparent shifting of the position of the object during refraction of light. Although the refractive index is the main factor of dependence in both the shifts, their expressions are different. The lateral shift has the expression $$D=t\times \dfrac{\sin (i-r)}{\cos r}$$ and the expression for the normal shift has been given in the solution.