The refractive index of the material of the prism for a monochromatic beam of light is 2 and its refracting angle is `60^(@)` . The angle of incidence corresponding to which this beam of light suffers minimum deviation is :

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Answer

To solve the problem of finding the angle of incidence corresponding to minimum deviation for a prism with a refractive index of 2 and a refracting angle of 60 degrees, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Refractive index of the prism, \( \mu = 2 \)
- Refracting angle of the prism, \( A = 60^\circ \)

2. **Use the Formula for Refractive Index**:
The formula relating the refractive index \( \mu \), the angle of the prism \( A \), and the minimum deviation \( \delta_m \) is given by:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substitute Known Values**:
Substitute \( A = 60^\circ \) into the formula:
\[ \mu = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{\sin\left(\frac{60^\circ}{2}\right)} \]
This simplifies to:
\[ \mu = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{\sin(30^\circ)} \]

4. **Calculate \( \sin(30^\circ) \)**:
We know that \( \sin(30^\circ) = \frac{1}{2} \). Thus, we can rewrite the equation:
\[ 2 = \frac{\sin\left(\frac{60^\circ + \delta_m}{2}\right)}{\frac{1}{2}} \]
This simplifies to:
\[ 2 = 2 \sin\left(\frac{60^\circ + \delta_m}{2}\right) \]
Therefore:
\[ 1 = \sin\left(\frac{60^\circ + \delta_m}{2}\right) \]

5. **Solve for \( \frac{60^\circ + \delta_m}{2} \)**:
The sine function equals 1 at \( 90^\circ \):
\[ \frac{60^\circ + \delta_m}{2} = 90^\circ \]
Multiplying both sides by 2 gives:
\[ 60^\circ + \delta_m = 180^\circ \]
Thus:
\[ \delta_m = 180^\circ - 60^\circ = 120^\circ \]

6. **Use the Minimum Deviation Formula**:
The relationship between the angle of incidence \( i \), the angle of the prism \( A \), and the minimum deviation \( \delta_m \) is given by:
\[ \delta_m = 2i - A \]
Substituting the known values:
\[ 120^\circ = 2i - 60^\circ \]

7. **Solve for \( i \)**:
Rearranging the equation gives:
\[ 2i = 120^\circ + 60^\circ = 180^\circ \]
Therefore:
\[ i = \frac{180^\circ}{2} = 90^\circ \]

### Final Answer:
The angle of incidence corresponding to which the beam of light suffers minimum deviation is \( i = 90^\circ \).

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