A crown glass prism of refracting angle `6^@` is to be used for deviation without dispersion with a flint glass of angle of prism `alpha` . Given : for crown glass `mu_r=1.513 and mu_v = 1.523,` for flint glass `mu_r=1.645 and mu_v=1.665`. Find `alpha`

Options

A. `3^@`
B. `4^@`
C. `4.5^@`
D. `5^@`

Answer

To solve the problem, we need to find the angle of the prism (α) for the flint glass that allows for deviation without dispersion when paired with a crown glass prism of refracting angle 6°.

### Step-by-Step Solution:

1. **Understand the Concept of Deviation without Dispersion**:
- Deviation without dispersion occurs when the angle of deviation for different wavelengths of light (red and violet) is the same. This means that the difference in the refractive indices multiplied by the angle of the prism should be equal for both glasses.

2. **Identify Given Values**:
- For crown glass:
- Angle of prism (A) = 6°
- Refractive index for red light (μ_r) = 1.513
- Refractive index for violet light (μ_v) = 1.523
- For flint glass:
- Refractive index for red light (μ'_r) = 1.645
- Refractive index for violet light (μ'_v) = 1.665
- Angle of prism (α) = ?

3. **Set Up the Equation for Deviation**:
- The condition for deviation without dispersion can be expressed as:
\[ (\mu_v - \mu_r) \cdot A = (\mu'_v - \mu'_r) \cdot \alpha \]

4. **Calculate the Differences in Refractive Indices**:
- For crown glass:
\[ \mu_v - \mu_r = 1.523 - 1.513 = 0.010 \]
- For flint glass:
\[ \mu'_v - \mu'_r = 1.665 - 1.645 = 0.020 \]

5. **Substitute Values into the Equation**:
- Substitute A = 6° into the equation:
\[ 0.010 \cdot 6 = 0.020 \cdot \alpha \]

6. **Solve for α**:
- Rearranging gives:
\[ \alpha = \frac{0.010 \cdot 6}{0.020} \]
- Simplifying:
\[ \alpha = \frac{0.060}{0.020} = 3° \]

7. **Final Answer**:
- The angle of the prism for flint glass (α) is **3°**.