Question

The dispersive powers of the materials of two lenses forming an achromatic combination are in the ratio of $4:3$. Effective focal length of the lenses is $+60\text{cm}$, then the focal lengths of the lenses should be

• A. $-2\text{cm},25\text{cm}$
• B. $20\text{cm},-25\text{cm}$
• C. $-15\text{cm},20\text{cm}$
• D. $15\text{cm},-20\text{cm}$

Answer 1

The correct option is D $15\text{cm},-20\text{cm}$

We know that for achromatic condition,
$\frac{{\omega }_1}{f_1}+\frac{{\omega }_2}{f_2}=0$
$\Rightarrow \frac{{\omega }_1}{{\omega }_2}=\frac{-f_1}{f_2}=\frac{4}{3}$
Also, $\frac{{\omega }_1}{f_1}+\frac{{\omega }_2}{f_2}=\frac{1}{f}$
$\Rightarrow 1+\frac{f_1}{f_2}=\frac{f_1}{f}$
$\Rightarrow 1-\frac{4}{3}=\frac{f_1}{60}$ [From (1)]
$\Rightarrow f_1=-20\text{cm}$
So, from (1)
$f_2=15\text{cm}$