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 \( \alpha \) for flint glass such that the net dispersion is zero when used with a crown glass prism of refracting angle \( 6^\circ \).

### Step-by-Step Solution:

1. **Understanding Dispersion**:
The dispersion of light through a prism is defined as the difference in the angle of deviation for different wavelengths of light. For our case, we want the net dispersion to be zero when using both prisms (crown glass and flint glass).

2. **Setting Up the Formula**:
The formula for the dispersion \( D \) can be expressed as:
\[ D = (\mu_v - \mu_r) \cdot A \]
where \( \mu_v \) is the refractive index for violet light, \( \mu_r \) is the refractive index for red light, and \( A \) is the angle of the prism.

3. **Dispersion for Crown Glass**:
For crown glass:
- \( \mu_{v,crown} = 1.523 \)
- \( \mu_{r,crown} = 1.513 \)
- Angle \( A = 6^\circ \)

The dispersion for crown glass \( D_{crown} \) is:
\[ D_{crown} = (1.523 - 1.513) \cdot 6^\circ = 0.01 \cdot 6^\circ = 0.06^\circ \]

4. **Dispersion for Flint Glass**:
For flint glass:
- \( \mu_{v,flint} = 1.665 \)
- \( \mu_{r,flint} = 1.645 \)
- Angle \( \alpha \) (unknown)

The dispersion for flint glass \( D_{flint} \) is:
\[ D_{flint} = (1.665 - 1.645) \cdot \alpha = 0.02 \cdot \alpha \]

5. **Setting the Condition for Zero Net Dispersion**:
For the net dispersion to be zero, the dispersion from the crown glass must equal the dispersion from the flint glass:
\[ D_{crown} = D_{flint} \]
Thus, we have:
\[ 0.06^\circ = 0.02 \cdot \alpha \]

6. **Solving for \( \alpha \)**:
Rearranging the equation to solve for \( \alpha \):
\[ \alpha = \frac{0.06^\circ}{0.02} = 3^\circ \]

### Final Answer:
The angle of the prism \( \alpha \) for flint glass is \( 3^\circ \).

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