A prism is made up of material of refractive index `sqrt3`. The angle of prism is `A`. If the angle of minimum deviation is equal to the angle of the prism, then the value of A is

Options

A. (a)`60^(@)`
B. (b)`50^(@)`
C. (c)`45^(@)`
D. (d)`30^(@)`

Answer

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 angle of the prism \( A \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The relationship between the refractive index \( \mu \), the angle of the prism \( A \), and the angle of minimum deviation \( \delta_m \) is given by the formula:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

2. **Substituting Given Values**:
Since \( \delta_m = A \), we can substitute \( \delta_m \) with \( A \) in the formula:
\[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substituting the Refractive Index**:
We know that \( \mu = \sqrt{3} \), so we can write:
\[ \sqrt{3} = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

4. **Using the Double Angle Formula**:
We can use the double angle formula for sine:
\[ \sin(A) = 2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right) \]
Substituting this into our equation gives:
\[ \sqrt{3} = \frac{2 \sin\left(\frac{A}{2}\right) \cos\left(\frac{A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

5. **Simplifying the Equation**:
The \( \sin\left(\frac{A}{2}\right) \) terms cancel out (assuming \( \sin\left(\frac{A}{2}\right) \neq 0 \)):
\[ \sqrt{3} = 2 \cos\left(\frac{A}{2}\right) \]

6. **Solving for Cosine**:
Rearranging gives:
\[ \cos\left(\frac{A}{2}\right) = \frac{\sqrt{3}}{2} \]

7. **Finding the Angle**:
The angle whose cosine is \( \frac{\sqrt{3}}{2} \) is \( 30^\circ \):
\[ \frac{A}{2} = 30^\circ \implies A = 60^\circ \]

8. **Conclusion**:
Therefore, the angle of the prism \( A \) is \( 60^\circ \).

### Final Answer:
The value of \( A \) is \( 60^\circ \).

---