Question

A ray of light travelling in a transparent medium of refractive index $ \mu $, falls on a surface separating the medium from air at an angle of incidence of 45$^{\circ}$. For which of the following value of $ \mu $. the ray can undergo total internal reflection?

• A. $\mu =1.33$
• B. $\mu =1.40$
• C. $\mu =1.50$
• D. $\mu =1.25$

Answer 1

The Correct Option is C

To determine when a ray of light can undergo total internal reflection, we need to apply the concept of critical angle. Total internal reflection occurs when a ray of light is traveling from a medium with a higher refractive index to a medium with a lower refractive index and strikes the interface at an angle greater than the critical angle.

The critical angle (\(\theta_c\)) can be calculated using the formula:  

\[\theta_c = \sin^{-1}\left(\frac{1}{\mu}\right)\]

 where \(\mu\) is the refractive index of the medium from which the light is coming towards the boundary with air.

Given that the angle of incidence is \(45^\circ\), for total internal reflection to occur, the condition should be: 

\[\theta_i = 45^\circ > \theta_c\]

We will now calculate the critical angle for each provided option:

1. For \(\mu = 1.33\): 

\[\theta_c = \sin^{-1}\left(\frac{1}{1.33}\right) \approx 48.75^\circ\]

1. For \(\mu = 1.40\): 

\[\theta_c = \sin^{-1}\left(\frac{1}{1.40}\right) \approx 45.58^\circ\]

1. For \(\mu = 1.50\): 

\[\theta_c = \sin^{-1}\left(\frac{1}{1.50}\right) \approx 41.81^\circ\]

1. For \(\mu = 1.25\): 

\[\theta_c = \sin^{-1}\left(\frac{1}{1.25}\right) \approx 53.13^\circ\]

Comparing these values, only when \(\mu = 1.50\) is the critical angle (\(41.81^\circ\)) less than \(45^\circ\). In this case, \(\theta_i = 45^\circ\) is greater than \(\theta_c\), allowing total internal reflection to occur.

Thus, the correct answer is \(\mu = 1.50\).