Question

A double convex lens is made of glass which has refractive index 1.55 for violet rays and 1.50 for red rays. If the focal length of violet rays is 20 cm, the focal length for red rays is :

• A. 9 cm
• B. 18 cm
• C. 20 cm
• D. 22 cm

Answer 1

Answer: (D)

22 cm

$\underset{–}{Given}:$

$\bullet$ $f_{{}_V}=20cm$

$\bullet$ ${\mu }_{{}_V}=1.55$

$\bullet$ ${\mu }_{{}_R}=1.50$

$\bullet$ Refractive index doesn't have a unit.

$To$ $find:$ $f_{{}_R}=$ $?$

$\underset{–}{Solution}:$

$\frac{1}{f}=(\mu -1)\times ({\frac{1}{R}}_1-{\frac{1}{R}}_2)$ Since lens is the same, $R_1$ and $R_2$ are common for both.

$\therefore {\frac{1}{f}}_{{}_V}=({\mu }_{{}_V}-1)\times ({\frac{1}{R}}_1-{\frac{1}{R}}_2)$

$\therefore {\frac{1}{f}}_{{}_R}=({\mu }_{{}_R}-1)\times ({\frac{1}{R}}_1-{\frac{1}{R}}_2)$

$\therefore \frac{\left({\frac{1}{f}}_{{}_V}\right)}{\left({\frac{1}{f}}_{{}_R}\right)}=\frac{({\mu }_{{}_V}-1)\times ({\frac{1}{R}}_1-{\frac{1}{R}}_2)}{({\mu }_{{}_R}-1)\times ({\frac{1}{R}}_1-{\frac{1}{R}}_2)}=\frac{({\mu }_{{}_V}-1)}{({\mu }_{{}_R}-1)}$

$\therefore$ ${\frac{1}{f}}_{{}_V}÷{\frac{1}{f}}_{{}_R}=$ $\frac{f_{{}_R}}{f_{{}_V}}=\frac{{\mu }_{{}_V}-1}{{\mu }_{{}_R}-1}$

$\therefore$ $f_{{}_R}=\frac{f_{{}_V}\times ({\mu }_{{}_V}-1)}{({\mu }_{{}_R}-1)}=\frac{20\times (1.55-1)}{(1.50-1)}=\frac{20\times 0.55}{0.5}=\frac{20\times 55}{50}=2\times 11=22$

$\underset{–}{Answer}:$ Focal length for red rays is 22cm.