A thin prism of angle `15^(@)` made of glass of refractive index `mu_(1)=1.5` is combined with another prism of glass of refractive index `mu_(2)=1.75`. The combination of the prism produces dispersion without deviation. The angle of the second prism should be

Options

A. `7^(@)`
B. `10^(@)`
C. `12^(@)`
D. `5^(@)`

Explanation

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

### Step 1: Understand the Problem
We have two prisms:
- Prism 1: Angle \( A_1 = 15^\circ \), Refractive index \( \mu_1 = 1.5 \)
- Prism 2: Angle \( A_2 \) (unknown), Refractive index \( \mu_2 = 1.75 \)

The combination of these two prisms produces zero total deviation.

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

### Step 3: Calculate Deviation for Each Prism
For Prism 1:
\[ \delta_1 = (\mu_1 - 1) \cdot A_1 = (1.5 - 1) \cdot 15^\circ = 0.5 \cdot 15^\circ = 7.5^\circ \]

For Prism 2, the deviation will be:
\[ \delta_2 = (\mu_2 - 1) \cdot A_2 = (1.75 - 1) \cdot A_2 = 0.75 \cdot A_2 \]

### Step 4: Set Up the Equation for Zero Total Deviation
Since the total deviation is zero, we can write:
\[ \delta_1 - \delta_2 = 0 \]
This means:
\[ \delta_1 = \delta_2 \]
Substituting the values we have:
\[ 7.5^\circ = 0.75 \cdot A_2 \]

### Step 5: Solve for \( A_2 \)
To find \( A_2 \), rearrange the equation:
\[ A_2 = \frac{7.5^\circ}{0.75} = 10^\circ \]

### Conclusion
The angle of the second prism \( A_2 \) is \( 10^\circ \).