question_answer

A ray of light moving in air strikes at origin at grazing angle and then follows a path \[2y={{x}^{2}}\]for \[x\ge 0.\] The correct variation of refractive index with x co-ordinate is:

A) \[\sqrt{1+4{{x}^{2}}}\]

B) \[\sqrt{1+2{{x}^{2}}}\]

C) \[\sqrt{1+{{x}^{2}}}\]

D) \[\sqrt{1+\frac{{{x}^{2}}}{2}}\]

Correct Answer: C

Solution :

Given, \[y=\frac{{{x}^{2}}}{2}\]

\[\therefore \]\[\frac{dy}{dx}=\frac{2x}{2}=x\]

Or \[\tan \theta =x\]

Now \[\frac{\sin 90{}^\circ }{\sin \theta }=\mu \]

Or \[\mu =\frac{1}{\left( \frac{1}{\sqrt{1+{{x}^{2}}}} \right)}=\sqrt{1+{{x}^{2}}}\]