A thin prism P with angle `4^(@)` and made from glass of refractive index 1.54 is combined with another thin prism P made from glass of refractive index 1.72 to produce dispersion without deviation The angle of prism P is

Options

A. `5.33^(@)`
B. `4^(@)`
C. `2.6^(@)`
D. `3^(@)`

Explanation

To solve the problem, we need to find the angle of the second prism (let's call it \( A_2 \)) that, when combined with the first prism (with angle \( A_1 = 4^\circ \) and refractive index \( \mu_1 = 1.54 \)), results in no net deviation. The second prism has a refractive index \( \mu_2 = 1.72 \).

### Step-by-Step Solution:

1. **Understand the Condition for No Deviation**:
- For two prisms combined to produce dispersion without deviation, the total deviation must be zero:
\[ \delta_1 + \delta_2 = 0 \]

2. **Use the Formula for Deviation of a Thin Prism**:
- The deviation \( \delta \) for a thin prism is given by:
\[ \delta = (\mu - 1) \cdot A \]
- For the first prism:
\[ \delta_1 = (\mu_1 - 1) \cdot A_1 = (1.54 - 1) \cdot 4^\circ \]
- For the second prism:
\[ \delta_2 = (\mu_2 - 1) \cdot A_2 = (1.72 - 1) \cdot A_2 \]

3. **Set Up the Equation**:
- Substitute the expressions for \( \delta_1 \) and \( \delta_2 \) into the no deviation condition:
\[ (1.54 - 1) \cdot 4 + (1.72 - 1) \cdot A_2 = 0 \]

4. **Calculate \( \delta_1 \)**:
- Calculate \( \delta_1 \):
\[ \delta_1 = 0.54 \cdot 4 = 2.16 \]

5. **Substitute and Solve for \( A_2 \)**:
- Now substitute \( \delta_1 \) into the equation:
\[ 2.16 + (0.72) \cdot A_2 = 0 \]
- Rearranging gives:
\[ (0.72) \cdot A_2 = -2.16 \]
- Thus,
\[ A_2 = \frac{-2.16}{0.72} \]
- Calculating \( A_2 \):
\[ A_2 = -3^\circ \]
- Since we are looking for a positive angle, we take the absolute value:
\[ A_2 = 3^\circ \]

6. **Conclusion**:
- The angle of the second prism \( A_2 \) is \( 3^\circ \).

### Final Answer:
The angle of prism P is \( 3^\circ \).