Question

For an achromatic combination, three different lenses are combined. Dispersive power of their materials are 0.066, 0.055 and 0.040 and their main focal lengths are - 22 cm, -11 cm and 'f ' respectively then find the value of ' f ' (in cm.)

Answer 1

$\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}+\frac{1}{f_3}\text{Differentiating}\Rightarrow \frac{-df}{f^2}=\frac{-d{f}_1}{f_1^2}-\frac{d{f}_2}{f_2^2}-\frac{d{f}_3}{f_2^2}$
$\frac{\omega }{f}=\frac{{\omega }_1}{f_1}+\frac{{\omega }_2}{f_2}+\frac{{\omega }_3}{f_3}$
For achromatic combination $\omega =0\Rightarrow \frac{-0.066}{22}-\frac{0.055}{11}+\frac{0.040}{f}=0$
$\Rightarrow -3-5+\frac{40}{f}=0$
$f=5\text{cm}$