The angle of minimum deviation for prism of angle `pi//3 is pi//6`. Calculate the velocity of light in the material of the prism if the velocity of light in vacuum is `3 xx 10^8 ms^-1`.

Options

A. `2.12 xx 10^(8) ms^(-1)`
B. `1.12 xx 10^(8) ms^(-1)`
C. `4.12 xx 10^(8) ms^(-1)`
D. `5.12 xx 10^(8) m s^(-1)`

Explanation

To solve the problem, we need to calculate the velocity of light in the material of a prism given the angle of the prism and the angle of minimum deviation. Here’s the step-by-step solution:

### Step 1: Identify the Given Values
- Angle of the prism \( A = \frac{\pi}{3} \) radians (which is 60 degrees).
- Angle of minimum deviation \( D = \frac{\pi}{6} \) radians (which is 30 degrees).
- Velocity of light in vacuum \( c = 3 \times 10^8 \, \text{m/s} \).

### Step 2: Calculate the Refractive Index \( \mu \)
The refractive index \( \mu \) of the prism can be calculated using the formula:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

### Step 3: Substitute the Values into the Formula
1. Calculate \( A + D \):
\[ A + D = \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]
2. Now calculate \( \frac{A + D}{2} \) and \( \frac{A}{2} \):
\[ \frac{A + D}{2} = \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4} \]
\[ \frac{A}{2} = \frac{\frac{\pi}{3}}{2} = \frac{\pi}{6} \]

### Step 4: Calculate the Sine Values
1. Calculate \( \sin\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{6}\right) \):
\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]

### Step 5: Substitute the Sine Values into the Refractive Index Formula
\[ \mu = \frac{\sin\left(\frac{\pi}{4}\right)}{\sin\left(\frac{\pi}{6}\right)} = \frac{\frac{\sqrt{2}}{2}}{\frac{1}{2}} = \sqrt{2} \]

### Step 6: Calculate the Velocity of Light in the Material of the Prism
The velocity of light in the material of the prism \( v \) can be calculated using the formula:
\[ v = \frac{c}{\mu} \]
Substituting the values:
\[ v = \frac{3 \times 10^8 \, \text{m/s}}{\sqrt{2}} \approx \frac{3 \times 10^8}{1.414} \approx 2.12 \times 10^8 \, \text{m/s} \]

### Final Answer
The velocity of light in the material of the prism is approximately \( 2.12 \times 10^8 \, \text{m/s} \).

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