Question

What is equivalent lens? Derive an expression for equivalent focal length of two lenses of focal lengths \(f_1\) and \(f_2\) kept at a distance \(d\).

Hint:

For two lenses at distance \(d\), the equivalent focal length is: \[ f_{\text{eq}} = \left( \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \right)^{-1} \]

Answer 1

Equivalent Lens:
An equivalent lens is formed when two or more lenses are placed in contact with each other or at a finite distance. The focal length of the equivalent lens is determined by the combination of the focal lengths of the individual lenses.
Expression for Equivalent Focal Length:
For two lenses of focal lengths \(f_1\) and \(f_2\) kept at a distance \(d\), the formula for the equivalent focal length \(f_{\text{eq}}\) can be derived using the lens formula for each lens and the condition for combined focal length. The combined power \(P_{\text{eq}}\) of two lenses is the sum of their individual powers: \[ P_{\text{eq}} = P_1 + P_2 \] Where: \[ P_1 = \frac{1}{f_1}, \quad P_2 = \frac{1}{f_2} \] Thus: \[ \frac{1}{f_{\text{eq}}} = \frac{1}{f_1} + \frac{1}{f_2} \] For lenses kept at a distance \(d\), the additional term due to the separation distance between the lenses must be included. The effective focal length is given by the relation: \[ \frac{1}{f_{\text{eq}}} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \] Therefore, the equivalent focal length of the two lenses placed at a distance \(d\) is: \[ f_{\text{eq}} = \left( \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2} \right)^{-1} \]