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^(@)`. For which of the following value of `mu` the ray can undergo total internal reflection ?

Options

A. `mu=1.33`
B. `mu=1.41`
C. `mu=1.50`
D. `mu=1.25`

Explanation

To determine the value of the refractive index \( \mu \) for which a ray of light can undergo total internal reflection when it travels from a transparent medium to air at an angle of incidence of \( 45^\circ \), we can follow these steps:

### Step 1: Understand the condition for total internal reflection
Total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index. The condition for total internal reflection is given by:
\[ \mu > \frac{1}{\sin \theta_C} \]
where \( \theta_C \) is the critical angle.

### Step 2: Determine the critical angle
The critical angle \( \theta_C \) can be found using Snell's law, which states:
\[ \mu \sin \theta_C = n \sin 90^\circ \]
Here, \( n \) is the refractive index of air, which is approximately 1. Therefore, we can simplify this to:
\[ \mu \sin \theta_C = 1 \]
From this, we can express \( \sin \theta_C \) as:
\[ \sin \theta_C = \frac{1}{\mu} \]

### Step 3: Set up the inequality for total internal reflection
Since we are given that the angle of incidence \( \theta \) is \( 45^\circ \), we can set \( \theta_C = 45^\circ \). Thus, we have:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \approx 0.707 \]
Substituting this into our inequality for total internal reflection gives:
\[ \mu > \frac{1}{\sin 45^\circ} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]

### Step 4: Calculate the numerical value
Calculating \( \sqrt{2} \):
\[ \sqrt{2} \approx 1.414 \]
Thus, the condition for total internal reflection becomes:
\[ \mu > 1.414 \]

### Step 5: Conclusion
For the ray to undergo total internal reflection, the refractive index \( \mu \) must be greater than \( 1.414 \). Therefore, the minimum value of \( \mu \) that satisfies this condition is \( 1.50 \).

### Final Answer
The value of \( \mu \) for which the ray can undergo total internal reflection is \( \mu = 1.50 \).

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