The dispersive powers of flint glass and crown glass are 0.053 and 0.034 respectively and their mean refractive indices are 1.68 and 1.53 for white light .Calculate the angle of the flint glass prism required to form an achromatic combination with a crown glass prism of refracting anlge `4^@` .

Options

A. `2^@`
B. `4^@`
C. `5^@`
D. `6^@`

Answer

To solve the problem, we need to find the angle of the flint glass prism (A_f) required to form an achromatic combination with a crown glass prism (A_c) of refracting angle 4 degrees.

### Step-by-Step Solution:

1. **Understand the Concept of Achromatic Combination**:
An achromatic combination occurs when the dispersion of light through two prisms cancels out. The condition for this is given by:
\[ (\mu_v - \mu_r) A_f = (\mu_{v_c} - \mu_{r_c}) A_c \]
where \( \mu_v \) and \( \mu_r \) are the refractive indices for violet and red light for flint glass, and \( \mu_{v_c} \) and \( \mu_{r_c} \) are the refractive indices for crown glass.

2. **Use Dispersive Powers**:
The dispersive power (ω) is defined as:
\[ \omega = \frac{\mu_v - \mu_r}{\mu - 1} \]
From this, we can express the difference in refractive indices for violet and red light:
\[ \mu_v - \mu_r = \omega (\mu - 1) \]

3. **Set Up the Equation**:
For flint glass:
\[ \mu_v - \mu_r = \omega_f (n_f - 1) \]
For crown glass:
\[ \mu_{v_c} - \mu_{r_c} = \omega_c (n_c - 1) \]

4. **Substitute into the Achromatic Condition**:
The achromatic condition becomes:
\[ \omega_f (n_f - 1) A_f = \omega_c (n_c - 1) A_c \]
Rearranging gives:
\[ A_f = \frac{\omega_c (n_c - 1) A_c}{\omega_f (n_f - 1)} \]

5. **Insert Known Values**:
- \( \omega_f = 0.053 \)
- \( \omega_c = 0.034 \)
- \( n_f = 1.68 \)
- \( n_c = 1.53 \)
- \( A_c = 4^\circ \)

Plugging these values into the equation:
\[ A_f = \frac{0.034 \times (1.53 - 1) \times 4}{0.053 \times (1.68 - 1)} \]

6. **Calculate Each Part**:
- \( n_c - 1 = 1.53 - 1 = 0.53 \)
- \( n_f - 1 = 1.68 - 1 = 0.68 \)

Thus:
\[ A_f = \frac{0.034 \times 0.53 \times 4}{0.053 \times 0.68} \]

7. **Perform the Calculation**:
- Calculate the numerator:
\[ 0.034 \times 0.53 \times 4 = 0.07204 \]
- Calculate the denominator:
\[ 0.053 \times 0.68 = 0.03604 \]
- Therefore:
\[ A_f = \frac{0.07204}{0.03604} \approx 2 \]

8. **Final Answer**:
The angle of the flint glass prism required to form an achromatic combination is approximately \( 2^\circ \).