The angle of minimum deviation for a glass prism with `mu=sqrt3` equals the refracting angle of the prism. What is the angle of the prism?

Options

A. `45^(@)`
B. `30^(@)`
C. `90^(@)`
D. `60^(@)`

Explanation

To solve the problem, we need to find the angle of the prism \( A \) given that the refractive index \( \mu = \sqrt{3} \) and the angle of minimum deviation \( \delta_m \) is equal to the refracting angle \( A \).

### Step-by-Step Solution:

1. **Understanding the relationship**: We know that for a prism, the refractive index \( \mu \) is given by the formula:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Given that \( \delta_m = A \), we can substitute this into the equation.

2. **Substituting the values**: Substituting \( \delta_m = A \) into the formula, we get:
\[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]
Since \( \mu = \sqrt{3} \), we can write:
\[ \sqrt{3} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

3. **Using the double angle identity**: We can use the identity \( \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \) to rewrite the equation:
\[ \sqrt{3} = \frac{2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
This simplifies to:
\[ \sqrt{3} = 2 \cos\left(\frac{A}{2}\right) \]

4. **Rearranging the equation**: Rearranging gives:
\[ \cos\left(\frac{A}{2}\right) = \frac{\sqrt{3}}{2} \]

5. **Finding the angle**: The angle whose cosine is \( \frac{\sqrt{3}}{2} \) is \( 30^\circ \). Therefore:
\[ \frac{A}{2} = 30^\circ \]
Multiplying both sides by 2 gives:
\[ A = 60^\circ \]

6. **Conclusion**: The angle of the prism is \( 60^\circ \).

### Final Answer:
The angle of the prism \( A \) is \( 60^\circ \).

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