Question

A ray of light travelling in air is incident on one face of a parallel glass slab of thickness \( t \) and refractive index \( \mu \) at an angle of incidence \( i \). Total time spent by the ray inside the slab is

• A. \( \frac{\mu^2 t}{c\sqrt{1 - \mu^2 \sin^2 i}} \)
• B. \( \frac{\mu t}{c\sqrt{\mu^2 - \sin^2 i}} \)
• C. \( \frac{\mu^2 t}{c\sqrt{\mu^2 - \sin^2 i}} \)
• D. \( \frac{t}{c\sqrt{\mu^2 - \sin^2 i}} \)

Hint:

To easily check the dimensional consistency of the options, notice that \( T \) must have the unit of time (\( \frac{\text{length}}{\text{velocity}} = \frac{t}{c} \)). Since \( \mu \) is a dimensionless refractive index, any expression where \( \frac{t}{c} \) is multiplied by dimensionless functions of \( \mu \) and \( \sin i \) is dimensionally correct. Thus, analyzing the boundary condition at normal incidence (\( i = 0 \)), the time must reduce to \( T = \frac{\mu t}{c} \). Substituting \( i = 0 \) into option (C) gives \( \frac{\mu^2 t}{c\sqrt{\mu^2}} = \frac{\mu t}{c} \), which confirms its correctness.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When light enters a medium of higher refractive index, it slows down and changes its path (refraction).
The time taken to cross the slab is the ratio of the distance traveled inside the slab to the speed of light in that medium.
The distance is the length of the refracted ray, and the speed is \(v = c/\mu\).
Step 2: Key Formula or Approach:
Snell's Law: \(1 \cdot \sin i = \mu \cdot \sin r\).
Distance traveled inside slab: \(d = \frac{t}{\cos r}\).
Speed inside slab: \(v = \frac{c}{\mu}\).
Time taken: \(T = \frac{d}{v}\).
Step 3: Detailed Explanation:
Step 1: Find the distance \(d\). From geometry, \(\cos r = \frac{t}{d} \implies d = \frac{t}{\cos r}\).
Step 2: Find the speed \(v\). \(v = \frac{c}{\mu}\).
Step 3: Calculate Time \(T\).
\[T = \frac{t/\cos r}{c/\mu} = \frac{\mu t}{c \cos r}\]
Step 4: Use Snell's Law to eliminate \(r\).
\[\sin r = \frac{\sin i}{\mu}\]
We know \(\cos r = \sqrt{1 - \sin^2 r} = \sqrt{1 - \left(\frac{\sin i}{\mu}\right)^2} = \sqrt{\frac{\mu^2 - \sin^2 i}{\mu^2}}\)
\[\cos r = \frac{\sqrt{\mu^2 - \sin^2 i}}{\mu}\]
Step 5: Substitute \(\cos r\) back into the time equation:
\[T = \frac{\mu t}{c \cdot \left(\frac{\sqrt{\mu^2 - \sin^2 i}}{\mu}\right)}\]
\[T = \frac{\mu^2 t}{c \sqrt{\mu^2 - \sin^2 i}}\]
This matches option (C).
Step 4: Final Answer:
The total time spent inside the glass slab is \(\frac{\mu^2 t}{c \sqrt{\mu^2 - \sin^2 i}}\).