There is a thin prism of angle `5^(@)` made of material having refractive index 1.6. There is another thin prism made of material of refractive index 1.8. These two prisms are combined in such a manner that they may produce dispersion without mean deviation. What should be the angle of second prism ?

Options

A. `10^(@)`
B. `3.75^(@)`
C. `4^(@)`
D. `6.75^(@)`

Explanation

To solve the problem of finding the angle of the second prism that produces dispersion without mean deviation when combined with the first prism, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two prisms. The first prism has an angle \( A_1 = 5^\circ \) and a refractive index \( n_1 = 1.6 \). The second prism has a refractive index \( n_2 = 1.8 \), and we need to find its angle \( A_2 \).

2. **Use the Deviation Formula**: For small angles (less than \( 10^\circ \)), the deviation \( \delta \) produced by a prism can be approximated using the formula:
\[ \delta = (n - 1) \cdot A \]
where \( n \) is the refractive index and \( A \) is the angle of the prism.

3. **Calculate Deviation for the First Prism**: For the first prism:
\[ \delta_1 = (n_1 - 1) \cdot A_1 = (1.6 - 1) \cdot 5^\circ = 0.6 \cdot 5^\circ = 3^\circ \]

4. **Set Up the Equation for the Second Prism**: Since the two prisms are combined to produce dispersion without mean deviation, the deviation produced by the second prism must be equal in magnitude but opposite in direction to that of the first prism. Therefore, we have:
\[ \delta_2 = (n_2 - 1) \cdot A_2 \]
Setting \( \delta_1 = \delta_2 \):
\[ 3^\circ = (1.8 - 1) \cdot A_2 \]

5. **Solve for \( A_2 \)**:
\[ 3^\circ = 0.8 \cdot A_2 \]
\[ A_2 = \frac{3^\circ}{0.8} = 3.75^\circ \]

6. **Conclusion**: The angle of the second prism \( A_2 \) is \( 3.75^\circ \).

### Final Answer:
The angle of the second prism should be \( 3.75^\circ \).