question_answer

The x-z plane separates two media A and B of refractive indices \[{{\mu }_{1}}=1.5\] and \[{{\mu }_{2}}=2\]. A ray of light travels from A to B. Its directions in the two media are given by unit vectors \[{{\vec{u}}_{1}}=a\hat{i}+b\hat{j}\] and \[{{\vec{u}}_{2}}=c\hat{i}+b\hat{j}\] then

A) \[\frac{a}{c}=\frac{4}{3}\]       

B) \[\frac{a}{c}=\frac{3}{4}\]

C) \[\frac{b}{d}=\frac{4}{3}\]       

D) \[\frac{b}{d}=\frac{3}{4}\]

Correct Answer: A

Solution :

[a] Applying Snell's law \[{{\mu }_{1}}\sin i={{\mu }_{2}}\sin r\] \[\left( \frac{3}{2} \right)\left( \frac{a}{\sqrt{{{a}^{2}}+{{b}^{2}}}} \right)=2\left( \frac{c}{\sqrt{{{c}^{2}}+{{d}^{2}}}} \right)\] Here, \[\sqrt{{{a}^{2}}+{{b}^{2}}}=\sqrt{{{c}^{2}}+{{d}^{2}}}=1,\,\,\,\,\,\therefore \,\,\,\,\,\frac{a}{c}=\frac{4}{3}\]