Which of the following spherical lenses does not exhibit dispersion? The radii of curvature of the surfaces of the lenses are given in diagrams.
Answer & explanation
Correct answer: option 3
The correct answer is Option (3)
A lens disperses white light because its focal length depends on the refractive index \( \mu \), which is different for different colours. By the lens maker's formula (thin lens)
\( \displaystyle \frac{1}{f} = (\mu - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \)
If the bracket is zero, the power is zero for every colour, whatever \( \mu \) is. Such a lens bends no colour, so it cannot separate the colours: it shows no dispersion.
The lens of Option (3) has both surfaces of the same radius \( R \), curved the same way, so \( R_1 = R_2 = R \):
\( \displaystyle \frac{1}{f} = (\mu - 1)\left(\frac{1}{R} - \frac{1}{R}\right) = 0 \)