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

Answer

To solve the problem, we need to find the angle of the second prism (let's denote it as \( A_2 \)) that, when combined with the first prism of angle \( A_1 = 15^\circ \) and refractive index \( \mu_1 = 1.5 \), results in dispersion without deviation. The refractive index of the second prism is \( \mu_2 = 1.75 \).

### 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:
\[ \frac{A_1}{A_2} = \frac{\mu_2 - 1}{\mu_1 - 1} \]
where \( A_1 \) is the angle of the first prism, \( A_2 \) is the angle of the second prism, \( \mu_1 \) is the refractive index of the first prism, and \( \mu_2 \) is the refractive index of the second prism.

2. **Substituting the Known Values**:
We know:
- \( A_1 = 15^\circ \)
- \( \mu_1 = 1.5 \)
- \( \mu_2 = 1.75 \)

Substitute these values into the equation:
\[ \frac{15^\circ}{A_2} = \frac{1.75 - 1}{1.5 - 1} \]

3. **Calculating the Right Side of the Equation**:
Calculate \( \mu_2 - 1 \) and \( \mu_1 - 1 \):
\[ \mu_2 - 1 = 1.75 - 1 = 0.75 \]
\[ \mu_1 - 1 = 1.5 - 1 = 0.5 \]
Now substitute these values into the equation:
\[ \frac{15^\circ}{A_2} = \frac{0.75}{0.5} \]

4. **Simplifying the Right Side**:
Simplify \( \frac{0.75}{0.5} \):
\[ \frac{0.75}{0.5} = 1.5 \]
Now we have:
\[ \frac{15^\circ}{A_2} = 1.5 \]

5. **Finding \( A_2 \)**:
Rearranging the equation to solve for \( A_2 \):
\[ A_2 = \frac{15^\circ}{1.5} \]
Calculate \( A_2 \):
\[ A_2 = 10^\circ \]

### Final Answer:
The angle of the second prism \( A_2 \) should be \( 10^\circ \).

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