When a ray of light falls on a prism of angle `48^(@)` , it suffers minimum deviation. Refractive index of the material of prism is `1.6`. The angle of incidence is nearly

Options

A. `16^(@)`
B. `48^(@)`
C. `30^(@)`
D. `41^(@)`

Answer

To solve the problem, we need to find the angle of incidence (i) when a ray of light passes through a prism with an angle of 48 degrees and a refractive index of 1.6, suffering minimum deviation.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Angle of the prism (A) = 48 degrees
- Refractive index (n) = 1.6

2. **Use the Formula for Minimum Deviation:**
The relationship between the refractive index, angle of incidence (i), angle of prism (A), and minimum deviation (Δm) is given by:
\[ n = \frac{\sin(\Delta m + \frac{A}{2})}{\sin(\frac{A}{2})} \]

3. **Calculate \(\frac{A}{2}\):**
\[ \frac{A}{2} = \frac{48}{2} = 24 \text{ degrees} \]

4. **Substitute Values into the Formula:**
Plugging in the values into the refractive index formula:
\[ 1.6 = \frac{\sin(\Delta m + 24)}{\sin(24)} \]

5. **Calculate \(\sin(24)\):**
Using a calculator, we find:
\[ \sin(24) \approx 0.4067 \]

6. **Rearranging the Equation:**
Rearranging the equation gives:
\[ \sin(\Delta m + 24) = 1.6 \times 0.4067 \approx 0.6507 \]

7. **Find \(\Delta m + 24\):**
Now, we need to find the angle whose sine is approximately 0.6507:
\[ \Delta m + 24 = \sin^{-1}(0.6507) \]
Using a calculator, we find:
\[ \Delta m + 24 \approx 40.514 \text{ degrees} \]

8. **Calculate \(\Delta m\):**
Now, subtract 24 degrees from both sides:
\[ \Delta m \approx 40.514 - 24 \approx 16.514 \text{ degrees} \]

9. **Use the Minimum Deviation Formula:**
The relationship for minimum deviation is:
\[ \Delta m = 2i - A \]
Rearranging gives:
\[ i = \frac{\Delta m + A}{2} \]

10. **Substituting Values:**
Substitute \(\Delta m\) and \(A\) into the equation:
\[ i = \frac{16.514 + 48}{2} \approx \frac{64.514}{2} \approx 32.257 \text{ degrees} \]

11. **Final Calculation:**
Since we are looking for the angle of incidence when the prism suffers minimum deviation, we can round it to the nearest whole number:
\[ i \approx 32 \text{ degrees} \]

### Conclusion:
The angle of incidence is nearly **32 degrees**.