A ray of monochromatic light suffers minimum deviation of `38^(@)` while passing through a prism of refracting angle `60^(@)` . Refracting index of the prism material is

Explanation

To find the refractive index of the prism material given the minimum deviation and the refracting angle of the prism, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between angles**:
The formula for minimum deviation (D) in a prism is given by:
\[ D = i + e - A \]
where \(i\) is the angle of incidence, \(e\) is the angle of emergence, and \(A\) is the refracting angle of the prism. At minimum deviation, \(i = e\).

2. **Set up the equation for minimum deviation**:
Since \(i = e\) at minimum deviation, we can write:
\[ D = 2i - A \]
Rearranging gives:
\[ 2i = D + A \quad \Rightarrow \quad i = \frac{D + A}{2} \]

3. **Substitute the known values**:
Given \(D = 38^\circ\) and \(A = 60^\circ\):
\[ i = \frac{38^\circ + 60^\circ}{2} = \frac{98^\circ}{2} = 49^\circ \]

4. **Calculate the angle of refraction**:
The angle of refraction \(r\) can be calculated using the relationship:
\[ r = \frac{A}{2} = \frac{60^\circ}{2} = 30^\circ \]

5. **Use Snell's Law to find the refractive index**:
According to Snell's Law:
\[ \mu = \frac{\sin i}{\sin r} \]
Substituting the values we found:
\[ \mu = \frac{\sin 49^\circ}{\sin 30^\circ} \]

6. **Calculate the sine values**:
We know that \(\sin 30^\circ = 0.5\). Now we need to calculate \(\sin 49^\circ\). Using a calculator or sine table:
\[ \sin 49^\circ \approx 0.7547 \]

7. **Substitute the sine values into the equation**:
\[ \mu = \frac{0.7547}{0.5} = 1.5094 \]

8. **Final answer**:
The refractive index of the prism material is approximately:
\[ \mu \approx 1.51 \]