Glass Slab Light Time Calculation

This solution calculates the time a light ray spends inside a glass slab.

Key Concepts Used:

• Snell's Law: Relates angles of incidence and refraction.
• Speed of Light in Medium: $v = \frac{c}{\mu}$
• Time Calculation: Time = Distance / Speed

Step-by-Step Derivation:

1. Apply Snell's Law:

   The relationship between the angle of incidence ($i$) in air and the angle of refraction ($r$) inside the glass slab ($\mu$) is given by:

   $ \sin i = \mu \sin r $

   Solving for $\sin r$:

   $ \sin r = \frac{\sin i}{\mu} $

2. Calculate the distance travelled inside the slab:

   Let the path length of the light ray inside the slab be $L$. The thickness of the slab is $t$. The angle the light ray makes with the normal inside the slab is $r$.

   Consider the right-angled triangle formed by the thickness $t$ (adjacent to angle $r$) and the path length $L$ (hypotenuse).

   $ \cos r = \frac{t}{L} $

   Therefore, the distance $L$ is:

   $ L = \frac{t}{\cos r} $

   We need $\cos r$. Using the identity $\sin^2 r + \cos^2 r = 1$:

   $ \cos r = \sqrt{1 - \sin^2 r} $

   Substitute $\sin r = \frac{\sin i}{\mu}$:

   $ \cos r = \sqrt{1 - \left(\frac{\sin i}{\mu}\right)^2} = \sqrt{\frac{\mu^2 - \sin^2 i}{\mu^2}} = \frac{\sqrt{\mu^2 - \sin^2 i}}{\mu} $

   Now substitute $\cos r$ back into the equation for $L$:

   $ L = \frac{t}{\frac{\sqrt{\mu^2 - \sin^2 i}}{\mu}} = \frac{\mu t}{\sqrt{\mu^2 - \sin^2 i}} $

3. Calculate the speed of light inside the slab:

   The speed of light ($v$) in the glass slab is:

   $ v = \frac{c}{\mu} $

4. Calculate the total time spent:

   The time ($T$) is the distance $L$ divided by the speed $v$:

   $ T = \frac{L}{v} = \frac{\left(\frac{\mu t}{\sqrt{\mu^2 - \sin^2 i}}\right)}{\left(\frac{c}{\mu}\right)} $

   $ T = \frac{\mu t}{\sqrt{\mu^2 - \sin^2 i}} \times \frac{\mu}{c} $

   $ T = \frac{\mu^2 t}{c\sqrt{\mu^2 - \sin^2 i}} $

The total time spent by the ray inside the slab is $\frac{\mu^2 t}{c\sqrt{\mu^2 - \sin^2 i}}$.