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

Options

A. `7.8^(@)`
B. `4.5^(@)`
C. `6^(@)`
D. `1.3^(@)`

Answer

To solve the problem, we need to find the angle of the second prism \( P_2 \) such that when combined with the first prism \( P_1 \), they produce dispersion without average deviation.

### Step-by-Step Solution:

1. **Understanding the Condition for No Average Deviation**:
The condition for two prisms to produce dispersion without average deviation is given by the formula:
\[ A_1(\mu_1 - 1) = A_2(\mu_2 - 1) \]
where \( A_1 \) and \( A_2 \) are the angles of prisms \( P_1 \) and \( P_2 \) respectively, and \( \mu_1 \) and \( \mu_2 \) are their respective refractive indices.

2. **Substituting the Known Values**:
For prism \( P_1 \):
- Angle \( A_1 = 6^\circ \)
- Refractive index \( \mu_1 = 1.54 \)

For prism \( P_2 \):
- Angle \( A_2 = \theta \) (this is what we need to find)
- Refractive index \( \mu_2 = 1.72 \)

Now substituting these values into the equation:
\[ 6(1.54 - 1) = \theta(1.72 - 1) \]

3. **Calculating the Left Side**:
Calculate \( 1.54 - 1 = 0.54 \):
\[ 6 \times 0.54 = 3.24 \]

4. **Calculating the Right Side**:
Calculate \( 1.72 - 1 = 0.72 \):
\[ \theta \times 0.72 \]

5. **Setting the Equation**:
Now we have:
\[ 3.24 = \theta \times 0.72 \]

6. **Solving for \( \theta \)**:
To find \( \theta \), rearrange the equation:
\[ \theta = \frac{3.24}{0.72} \]

7. **Calculating \( \theta \)**:
Performing the division:
\[ \theta = 4.5^\circ \]

### Final Answer:
The angle of prism \( P_2 \) is \( \theta = 4.5^\circ \).

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