A prism of refractive index `m` and angle `A` is placed in the minimum deviation position. If the angle of minimum deviation is `A`, then the value of `A` in terms of `m` is

Options

A. (a)`sin^(-1)mu/2`
B. (b)`sin^(-1) sqrt((mu-1)/2)`
C. (c)`2 cos^(-1)(mu/2)`
D. (d)`cos^(-1)(mu/2)`

Answer

To solve the problem, we need to find the angle of minimum deviation \( A \) in terms of the refractive index \( \mu \) of the prism.

### Step-by-Step Solution:

1. **Understanding the Relationship**:
The refractive index \( \mu \) of a prism can be expressed in terms of the angle of the prism \( A \) and the angle of minimum deviation \( D \) using the formula:
\[ \mu = \frac{\sin\left(\frac{A + D}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]
Here, \( D \) is the angle of minimum deviation.

2. **Substituting the Given Information**:
According to the problem, the angle of minimum deviation \( D \) is equal to the angle of the prism \( A \). Therefore, we can substitute \( D = A \) into the equation:
\[ \mu = \frac{\sin\left(\frac{A + A}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Simplifying the Equation**:
This simplifies to:
\[ \mu = \frac{\sin(A)}{\sin\left(\frac{A}{2}\right)} \]

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

5. **Cancelling Terms**:
The \( \sin\left(\frac{A}{2}\right) \) terms cancel out:
\[ \mu = 2 \cos\left(\frac{A}{2}\right) \]

6. **Solving for \( A \)**:
To express \( A \) in terms of \( \mu \), we rearrange the equation:
\[ \cos\left(\frac{A}{2}\right) = \frac{\mu}{2} \]
Taking the inverse cosine gives:
\[ \frac{A}{2} = \cos^{-1}\left(\frac{\mu}{2}\right) \]
Therefore, multiplying both sides by 2:
\[ A = 2 \cos^{-1}\left(\frac{\mu}{2}\right) \]

### Final Answer:
Thus, the angle \( A \) in terms of the refractive index \( \mu \) is:
\[ A = 2 \cos^{-1}\left(\frac{\mu}{2}\right) \]