A glass prism A deviates the red and blue rays through `10^(@)` and `12^(@)` respectively . A second prism (B) deviates them through `8^(@)` and `10^(@)` respectively . What is the ratio of their dispersive powers ?

Options

A. `4 : 5`
B. `9 : 11`
C. `11 : 9`
D. `9 : 13`

Explanation

To find the ratio of the dispersive powers of two prisms A and B, we can follow these steps:

### Step 1: Understand the Definitions
Dispersive power (ω) of a prism is defined as the angular dispersion of light divided by the mean deviation. The angular dispersion is the difference in deviation between the blue and red light rays.

### Step 2: Calculate Angular Dispersion for Prism A
For prism A:
- Deviation of blue light (D_b) = 12°
- Deviation of red light (D_r) = 10°

Angular dispersion (ΔA) for prism A is given by:
\[ \Delta A = D_b - D_r = 12° - 10° = 2° \]

### Step 3: Calculate Mean Deviation for Prism A
Mean deviation (M_A) for prism A is the average of the deviations:
\[ M_A = \frac{D_b + D_r}{2} = \frac{12° + 10°}{2} = \frac{22°}{2} = 11° \]

### Step 4: Calculate Dispersive Power for Prism A
Dispersive power (ω_A) for prism A is given by:
\[ \omega_A = \frac{\Delta A}{M_A} = \frac{2°}{11°} \]

### Step 5: Calculate Angular Dispersion for Prism B
For prism B:
- Deviation of blue light (D_b) = 10°
- Deviation of red light (D_r) = 8°

Angular dispersion (ΔB) for prism B is given by:
\[ \Delta B = D_b - D_r = 10° - 8° = 2° \]

### Step 6: Calculate Mean Deviation for Prism B
Mean deviation (M_B) for prism B is the average of the deviations:
\[ M_B = \frac{D_b + D_r}{2} = \frac{10° + 8°}{2} = \frac{18°}{2} = 9° \]

### Step 7: Calculate Dispersive Power for Prism B
Dispersive power (ω_B) for prism B is given by:
\[ \omega_B = \frac{\Delta B}{M_B} = \frac{2°}{9°} \]

### Step 8: Calculate the Ratio of Dispersive Powers
Now, we can find the ratio of the dispersive powers of prism A to prism B:
\[ \text{Ratio} = \frac{\omega_A}{\omega_B} = \frac{\frac{2}{11}}{\frac{2}{9}} = \frac{2}{11} \times \frac{9}{2} = \frac{9}{11} \]

### Final Answer
The ratio of the dispersive powers of prism A to prism B is:
\[ 9 : 11 \]

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