A prism of a certain angle deviates the red and blue rays by `8^@` and `12^@` respectively. Another prism of the same angle deviates the red and blue rays by `10^@` and `14^@` respectively. The prisms are small angled and made of different materials. The dispersive powers of the materials of the prisms are in the ratio

Options

A. `5:6`
B. `9:11`
C. `6:5`
D. `11:9`

Answer

To solve the problem, we need to find the dispersive powers of two prisms made of different materials based on the given deviations for red and blue rays.

### Step-by-Step Solution:

1. **Understand the Formula for Dispersive Power**:
The dispersive power (ω) of a prism is given by the formula:
\[ \omega = \frac{\Delta_B - \Delta_R}{\Delta_{avg}} \]
where:
- \(\Delta_B\) = deviation for blue light
- \(\Delta_R\) = deviation for red light
- \(\Delta_{avg}\) = average deviation

2. **Calculate for the First Prism**:
- Given deviations:
- \(\Delta_B = 12^\circ\) (for blue)
- \(\Delta_R = 8^\circ\) (for red)
- Calculate the average deviation:
\[ \Delta_{avg} = \frac{\Delta_B + \Delta_R}{2} = \frac{12 + 8}{2} = 10^\circ \]
- Now, substitute into the dispersive power formula:
\[ \omega_1 = \frac{12 - 8}{10} = \frac{4}{10} = \frac{2}{5} \]

3. **Calculate for the Second Prism**:
- Given deviations:
- \(\Delta_B = 14^\circ\) (for blue)
- \(\Delta_R = 10^\circ\) (for red)
- Calculate the average deviation:
\[ \Delta_{avg} = \frac{\Delta_B + \Delta_R}{2} = \frac{14 + 10}{2} = 12^\circ \]
- Now, substitute into the dispersive power formula:
\[ \omega_2 = \frac{14 - 10}{12} = \frac{4}{12} = \frac{1}{3} \]

4. **Find the Ratio of Dispersive Powers**:
- Now we need to find the ratio of the dispersive powers of the two prisms:
\[ \frac{\omega_1}{\omega_2} = \frac{\frac{2}{5}}{\frac{1}{3}} = \frac{2}{5} \times \frac{3}{1} = \frac{6}{5} \]
- Therefore, the ratio of the dispersive powers is:
\[ \omega_1 : \omega_2 = 6 : 5 \]

### Final Answer:
The dispersive powers of the materials of the prisms are in the ratio \(6 : 5\).

---