Question

A deviation of$2°$ is produced in the yellow ray when the prism of crown and flint glass are chromatically combined. Taking dispersive powers of crown and flint glass as $0.02$ 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 _______$°$ (in $degree$). (Round off to the Nearest Integer)

Answer 1

Step 1. Given data

deviation,$\delta =2°$

Dispersive power of crown glass,$n_c=0.02$

Dispersive power of flint glass,$n_f=0.03$

Refractive index of crown glass,${\mu }_c=1.5$

Refractive index of flint glass,${\mu }_f=1.6$

Step 2. Solving for refracting angle

Since the two prisms are chromatically combined,

the net refraction angle is, ${\theta }_{net}=0$

${\theta }_c-{\theta }_f=0$

${\theta }_c={\theta }_f$

Where ${\theta }_c$ is refracting angle made by crown glass prism and ${\theta }_f$ is refracting angle made by flint glass prism

The refraction angle of prism equals dispersive power times the angle of deviation of prism,

$\Rightarrow n_c{\delta }_c=n_f{\delta }_f$

Where ${\delta }_c$ is the angle of deviation of crown glass prism and ${\delta }_f$ is angle of deviation of flint glass prism

The angle of deviation on combining the two prism is,

$\delta =2°={\delta }_c-{\delta }_f$

$\Rightarrow ={\delta }_c-\frac{n_c{\delta }_c}{n_f}=2°$

$\Rightarrow =\left(1-\frac{n_c}{n_f}\right){\delta }_c=2°$

$\Rightarrow =\left(1-\frac{0.02}{0.03}\right){\delta }_c=2°$

$\Rightarrow {\delta }_c=6°$

For Refracting angle $A_c$,

$\Rightarrow {\delta }_c=\left({\mu }_c-1\right)A_c$

$\Rightarrow 6=\left(1.5-1\right)A_c$

$\Rightarrow A_c=12$

Hence, the correct answer is $12$$°$.