Question

An achromatic convergent doublet of two lenses in contact has a power of +5D. The power of the converging lens is +6D. The ratio of the dispersive power of the converging and divergent lenses is:

• A. 3 : 7
• B. 2 : 3
• C. 1 : 5
• D. 5 : 3

Hint:

In an achromatic doublet, the power of the doublet is the sum of the powers of the individual lenses. The dispersive power ratio depends on the ratio of their powers.

Answer 1

The Correct Option is C

The condition of achromatism is \( W_1 P_1 + W_2 P_2 = 0 \) \[ \Rightarrow W_1 P_1 = -W_2 P_2 \] \[ \Rightarrow \frac{W_1}{W_2} = \frac{P_2}{P_1} \quad \dots {(i)} \] Now, \[ P_1 + P_2 = 4D \quad \dots {(ii)} \] but, Power of converting lens, \[ P_1 = 5D \] \(\therefore\) Power of diverging lens \[ P_2 = 4D - P_1 \quad {[From (ii)]} \] \[ = 4D - 5D = -D \] \(\therefore\) From Eq. (i), we have \[ \frac{W_1}{W_2} = \frac{P_2}{P_1} = \frac{-(-D)}{5D} = \frac{1}{5} \Rightarrow W_1 = \frac{1}{5} W_2 \]