Question

A deviation of $2^{\circ }$ is produced in the yellow ray when prism of crown and flint glass are achromatically combined. Taking dispersive powers of crown and flint glass are $0.02$ and $0.03$ respectively and refractive index for yellow light for these glasses are $1.5$ and $1.6$ respectively. The refracting angles for crown glass prism will be _______ ${}^{\circ }$(in degree) (Round off to the Nearest Integer)

Answer 1

Answer: 12

${\omega }_1=0.02;{\mu }_1=1.5;{\omega }_2=0.03;{\mu }_2=1.6$
Achromatic combination
$\therefore {\theta }_{\text{net }}=0$
${\theta }_1-{\theta }_2=0$
${\theta }_1={\theta }_2$
${\omega }_1{\delta }_1={\omega }_2{\delta }_2$
$\&{\delta }_{\text{net }}={\delta }_1-{\delta }_2=2^{\circ }$
${\delta }_1-\frac{{\omega }_1{\delta }_1}{{\omega }_2}=2^{\circ }$
${\delta }_1\left(1-\frac{{\omega }_1}{{\omega }_2}\right)=2^{\circ }$
${\delta }_1\left(1-\frac{2}{3}\right)=2^{\circ }$
${\delta }_1=6^{\circ }$
${\delta }_1=\left({\mu }_1-1\right)A_1$
$6^{\circ }=(1.5-1)A_1$
$A_1=12^{\circ }$