Question

A thin plano-concave lens with its curved face of radius of curvature R is made of glass of refractive index \( n_1 \). It is placed coaxially in contact with a thin equiconvex lens of same radius of curvature of refractive index \( n_2 \). Obtain the power of the combination lens.

Hint:

The total power of a combination of lenses in contact is the sum of the individual powers, and for curved surfaces, the focal length depends on the radius of curvature and the refractive index.

Answer 1

For a combination of lenses in contact, the effective power \( P \) is the sum of the powers of the individual lenses: \[ P_{\text{effective}} = P_1 + P_2 \] where \( P_1 \) is the power of the plano-concave lens and \( P_2 \) is the power of the equiconvex lens. The power \( P \) of a lens is given by: \[ P = \frac{1}{f} \] where \( f \) is the focal length of the lens. For a plano-concave lens with radius of curvature \( R \) and refractive index \( n_1 \): \[ \frac{1}{f_1} = \frac{n_1 - 1}{R} \] For the equiconvex lens with radius of curvature \( R \) and refractive index \( n_2 \): \[ \frac{1}{f_2} = \frac{n_2 - 1}{R} \] Now, the total power of the combination is: \[ P_{\text{effective}} = \frac{1}{f_1} + \frac{1}{f_2} = \frac{n_1 - 1}{R} + \frac{n_2 - 1}{R} \] \[ P_{\text{effective}} = \frac{(n_1 - 1) + (n_2 - 1)}{R} \] Thus, the effective power of the combination is: \[ P_{\text{effective}} = \frac{(n_1 + n_2 - 2)}{R} \]