Question

The angle of prism is A and one of its refracting surface is silvered. Light rays falling at an angle of incidence '2A' on the first surface return back through the same path after suffering reflection at the silvered surface. The refractive index of the material of the prism is

• A. $2 \sin\left(\frac{A}{2}\right)$
• B. $2 \tan A$
• C. $2 \cos A$
• D. $2 \sin A$

Hint:

Logic Tip: The phrase "returns back through the same path" or "retraces its path" is a classic physics code phrase meaning the ray hit the mirror surface exactly along the normal line. Instantly set the angle of incidence/reflection at that boundary to zero!

Answer 1

The Correct Option is C

Concept:
For a light ray to retrace its original path after reflecting from a surface (the silvered second face of the prism), it must strike that surface normally (perpendicularly). This means the angle of incidence at the second surface is $0^\circ$. We can then use the prism geometry equations ($A = r_1 + r_2$) and Snell's Law to find the refractive index.
Step 1: Identify the angles at the two surfaces.
Let the angle of incidence at the first surface be $i = 2A$. Let the angle of refraction at the first surface be $r_1$. Let the angle of incidence at the second (silvered) surface be $r_2$. Because the ray retraces its path, it must hit the silvered surface at exactly $90^\circ$ to the surface, which means the angle with the normal is zero: $$r_2 = 0^\circ$$
Step 2: Find the angle of refraction $r_1$.
The refracting angle of a prism ($A$) is related to the internal angles by: $$A = r_1 + r_2$$ Substitute $r_2 = 0$: $$A = r_1 + 0 \implies r_1 = A$$
Step 3: Apply Snell's Law at the first surface.
Snell's Law states $\mu_1 \sin i = \mu_2 \sin r$. Assuming the prism is in the air ($\mu_1 = 1$), the refractive index of the prism ($\mu$) is: $$\mu = \frac{\sin i}{\sin r_1}$$ Substitute $i = 2A$ and $r_1 = A$: $$\mu = \frac{\sin(2A)}{\sin A}$$
Step 4: Simplify using trigonometric identities.
Use the double angle identity $\sin(2A) = 2 \sin A \cos A$: $$\mu = \frac{2 \sin A \cos A}{\sin A}$$ Cancel out $\sin A$: $$\mu = 2 \cos A$$