When a ray of light enters from one medium to another then its velocity in second medium becomes doubled. The maximum value of angle of incidence so that total internal reflection may not take place will be

Options

A. `60^@`
B. `90^@`
C. `30^@`
D. `180^@`

Explanation

To solve the problem step by step, we need to determine the maximum angle of incidence for which total internal reflection does not occur when light travels from one medium to another, given that the velocity of light in the second medium is doubled.

### Step 1: Understand the relationship between velocity and refractive index
The velocity of light in a medium is given by the formula:
\[ v = \frac{c}{n} \]
where \(c\) is the speed of light in vacuum, and \(n\) is the refractive index of the medium.

### Step 2: Establish the relationship between the two media
Let’s denote the refractive indices of the first and second media as \(n_1\) and \(n_2\) respectively. According to the problem, when light enters the second medium, its velocity becomes double that in the first medium. Therefore, we can write:
\[ v_2 = 2v_1 \]
Substituting the expressions for velocity, we have:
\[ \frac{c}{n_2} = 2 \left(\frac{c}{n_1}\right) \]

### Step 3: Simplify the equation
From the equation above, we can simplify it:
\[ \frac{c}{n_2} = \frac{2c}{n_1} \]
Cancelling \(c\) from both sides gives:
\[ \frac{1}{n_2} = \frac{2}{n_1} \]
Cross-multiplying yields:
\[ n_2 = \frac{n_1}{2} \]

### Step 4: Apply Snell's Law 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 critical angle \(\theta_c\) can be found using Snell's law:
\[ \sin(\theta_c) = \frac{n_2}{n_1} \]
Substituting the value of \(n_2\):
\[ \sin(\theta_c) = \frac{\frac{n_1}{2}}{n_1} = \frac{1}{2} \]

### Step 5: Calculate the critical angle
To find the critical angle \(\theta_c\):
\[ \theta_c = \sin^{-1}\left(\frac{1}{2}\right) \]
This gives:
\[ \theta_c = 30^\circ \]

### Step 6: Conclusion
The maximum angle of incidence for which total internal reflection does not take place is equal to the critical angle, which we have calculated to be:
\[ \theta_c = 30^\circ \]

### Final Answer
The maximum value of the angle of incidence so that total internal reflection may not take place is \(30^\circ\).

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