Flint glass prism is joined by a crown glass prism to produce dispersion without deviation. The refractive indices of these for mean rays are `1.602` and `1.500` respectively. Angle of prism of flint prism is `10^(@)`, then the angle of prism for crown prism will be

Options

A. (a)`12^@2.4^(')`
B. (b)`12^@4^(')`
C. (c)`1.24^@`
D. (d)`12^@`

Answer

To solve the problem, we need to find the angle of the crown glass prism when it is joined with a flint glass prism to produce dispersion without deviation. Given the refractive indices and the angle of the flint prism, we can use the formula for deviation in prisms.

### Step-by-Step Solution:

1. **Understand the Condition of No Deviation**:
- For dispersion without deviation, the deviation produced by both prisms must be equal.
- This means that the deviation for the flint glass prism (D_f) is equal to the deviation for the crown glass prism (D_c).

2. **Use the Deviation Formula**:
- The formula for the deviation produced by a prism is given by:
\[ D = (\mu - 1) \times A \]
- Where \( \mu \) is the refractive index of the prism and \( A \) is the angle of the prism.

3. **Set Up the Equation**:
- For the flint glass prism:
\[ D_f = (\mu_f - 1) \times A_f \]
- For the crown glass prism:
\[ D_c = (\mu_c - 1) \times A_c \]
- Since \( D_f = D_c \):
\[ (\mu_f - 1) \times A_f = (\mu_c - 1) \times A_c \]

4. **Substitute the Known Values**:
- Given:
- \( \mu_f = 1.602 \) (for flint glass)
- \( \mu_c = 1.500 \) (for crown glass)
- \( A_f = 10^\circ \) (angle of flint prism)
- Substitute these values into the equation:
\[ (1.602 - 1) \times 10 = (1.500 - 1) \times A_c \]
- This simplifies to:
\[ 0.602 \times 10 = 0.500 \times A_c \]

5. **Solve for \( A_c \)**:
- Rearranging gives:
\[ A_c = \frac{0.602 \times 10}{0.500} \]
- Calculate the right-hand side:
\[ A_c = \frac{6.02}{0.500} = 12.04 \]

6. **Convert to Degrees and Minutes**:
- The angle \( A_c = 12.04^\circ \) can be expressed as:
- \( 12^\circ \) and \( 0.04 \times 60 \) minutes.
- \( 0.04 \times 60 = 2.4 \) minutes.
- Therefore, \( A_c = 12^\circ 2.4' \).

### Final Answer:
The angle of the crown glass prism is \( 12^\circ 2.4' \).