Question

A convex lens of focal length $5\text{cm}$ is used in the combination with a concave lens of unknown focal length to form an achromatic combination. If the dispersive powers of the lenses are in the ratio $2:3$ respectively. Then the value of focal length of concave lens will be :

• A. $\frac{15}{2}\text{cm}$
• B. $\frac{10}{3}\text{cm}$
• C. $\frac{6}{5}\text{cm}$
• D. $\frac{7}{3}\text{cm}$

Answer 1

The correct option is A $\frac{15}{2}\text{cm}$

Given :
Focal length of convex lens $\left(f_1\right)=5\text{cm}$

Ratio of dispersive powers of the lenses $\left(\frac{{\omega }_1}{{\omega }_2}\right)=2:3$

Let the focal length of the concave lens is $f_2.$

We know that for the creation of achromatic combination :

$\frac{{\omega }_1}{f_1}+\frac{{\omega }_2}{f_2}=0$

$\Rightarrow \frac{{\omega }_1}{f_1}=-\frac{{\omega }_2}{f_2}$

$\Rightarrow f_2=-f_1\frac{{\omega }_2}{{\omega }_1}$

From the given data :

$\Rightarrow f_2=-5\times \frac{3}{2}$

$\Rightarrow f_2=-\frac{15}{2}\text{cm}$

Hence, option $\left(a\right)$ is the correct answer.