A thin prism `P_(1)` with angle`6^(@)` and made from glass of refractive index 1.54 is combined with another thin prism `P_(2)` of refractive index 1.72 to produce dispersion without deviation. The angle of prism `P_(2)` will be

Options

A. `5^(@)24'`
B. `4^(@)30'`
C. `6^(@)`
D. `8^(@)`

Explanation

To solve the problem of finding the angle of prism \( P_2 \) that, when combined with prism \( P_1 \), produces dispersion without deviation, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Condition for Dispersion without Deviation**:
- For dispersion without deviation, the total deviation caused by both prisms must be zero. This means that the deviation caused by prism \( P_1 \) must equal the negative of the deviation caused by prism \( P_2 \).

2. **Formula for Deviation**:
- The deviation \( D \) for a thin prism is given by the formula:
\[ D = (\mu - 1) \cdot A \]
where \( \mu \) is the refractive index of the prism and \( A \) is the angle of the prism.

3. **Setting Up the Equation**:
- Let \( \mu_1 = 1.54 \) (for prism \( P_1 \)), \( A_1 = 6^\circ \) (angle of prism \( P_1 \)), \( \mu_2 = 1.72 \) (for prism \( P_2 \)), and \( A_2 \) be the angle of prism \( P_2 \) that we need to find.
- According to our condition:
\[ D_1 + D_2 = 0 \]
- This leads to:
\[ (\mu_1 - 1) \cdot A_1 = (\mu_2 - 1) \cdot A_2 \]

4. **Substituting Known Values**:
- Substitute the values into the equation:
\[ (1.54 - 1) \cdot 6 = (1.72 - 1) \cdot A_2 \]
- Simplifying gives:
\[ 0.54 \cdot 6 = 0.72 \cdot A_2 \]

5. **Calculating \( A_2 \)**:
- Calculate the left-hand side:
\[ 0.54 \cdot 6 = 3.24 \]
- Now, we can solve for \( A_2 \):
\[ 3.24 = 0.72 \cdot A_2 \]
\[ A_2 = \frac{3.24}{0.72} = 4.5^\circ \]

6. **Final Result**:
- The angle of prism \( P_2 \) is:
\[ A_2 = 4.5^\circ \text{ or } 4^\circ 30' \]

### Conclusion:
The angle of prism \( P_2 \) that produces dispersion without deviation when combined with prism \( P_1 \) is \( 4.5^\circ \).