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 two lenses is $+60cm$ then the focal lengths of the lenses should be:

• A. $-20$cm, $25$cm
• B. $20$cm, $-25$cm
• C. $-15$cm, $20$cm
• D. $15$cm, $-20$cm

Answer 1

Answer: (D)

$15$cm, $-20$cm

$P=P_1+P_2=\frac{1}{f_1}+\frac{1}{f_2}$

$\Rightarrow \frac{1}{60}=\frac{1}{f_1}+\frac{1}{f_2}$

$\Rightarrow \frac{f_1+f_2}{f_1{f}_2}=\frac{1}{60}$

Given that: $\frac{{\omega }_1}{{\omega }_2}=-\frac{f_1}{f_2}$

$\Rightarrow \frac{f_1}{f_2}=-\frac{4}{3}$

$\Rightarrow f_1=-\frac{4}{3}{f}_2$

Substituting in the above equation:

$\Rightarrow \frac{1}{60}=\frac{-\frac{4}{3}{f}_2+f_2}{-f_2^2\frac{4}{3}}$

$\Rightarrow \frac{1}{60}=\frac{-\frac{1}{3}}{-\frac{4}{3}{f}_2}$

$\Rightarrow f_2=15cm$

$\Rightarrow f_1=-\frac{4}{3}\times 15=-20cm$