A prism of prism angle \( A \) has one of its refracting surfaces silvered. Light rays falling at an angle of incidence \( 3A \) on the first surface return back through the same path after suffering reflection at the second (silvered) surface. The refractive index of the material of prism is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \mu = 3 - 4\sin^{2}A \)
A ray retraces its path after reflection at the silvered face only if it strikes that face normally. So inside the prism the refracted ray is perpendicular to the second face, and the geometry of the prism gives the angle of refraction at the first face.
\( r_{1} + r_{2} = A, \quad r_{2} = 0 \Rightarrow r_{1} = A \)
By Snell’s law at the first face (\( i = 3A \)):
\( \displaystyle \mu = \frac{\sin i}{\sin r_{1}} = \frac{\sin3A}{\sin A} \)
Using \( \sin3A = 3\sin A - 4\sin^{3}A \):
\( \displaystyle \mu = \frac{3\sin A - 4\sin^{3}A}{\sin A} = 3 - 4\sin^{2}A \)