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

Explanation

To solve the problem, we need to find the angle of the second prism (let's call it P2) that, when combined with the first prism (P1), produces dispersion without deviation.

### Step-by-Step Solution:

1. **Understanding the Condition for Dispersion without Deviation**:
The condition for two prisms to produce dispersion without deviation is given by the formula:
\[ (\mu_1 - 1) \cdot A_1 = (\mu_2 - 1) \cdot A_2 \]
where \( \mu_1 \) and \( \mu_2 \) are the refractive indices of the two prisms, and \( A_1 \) and \( A_2 \) are their respective angles.

2. **Identifying Given Values**:
- For the first prism (P1):
- Angle \( A_1 = 4^\circ \)
- Refractive index \( \mu_1 = 1.54 \)
- For the second prism (P2):
- Refractive index \( \mu_2 = 1.72 \)
- Angle \( A_2 \) is what we need to find.

3. **Substituting Values into the Formula**:
Substitute the known values into the equation:
\[ (1.54 - 1) \cdot 4 = (1.72 - 1) \cdot A_2 \]

4. **Calculating the Left Side**:
Calculate \( (1.54 - 1) \):
\[ 0.54 \cdot 4 = 2.16 \]

5. **Calculating the Right Side**:
Calculate \( (1.72 - 1) \):
\[ 0.72 \cdot A_2 \]

6. **Setting the Equation**:
Now we have:
\[ 2.16 = 0.72 \cdot A_2 \]

7. **Solving for \( A_2 \)**:
Rearranging the equation to find \( A_2 \):
\[ A_2 = \frac{2.16}{0.72} \]

8. **Calculating \( A_2 \)**:
Performing the division:
\[ A_2 = 3^\circ \]

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

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

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