In an achromatic combination of two prisms, the ratio of the mean deviations produced by V the two prisms is 2:3, the ratio of their dispersive power is

Options

A. `3:2`
B. `3:2`
C. `1:1`
D. `4:9`

Explanation

To solve the problem regarding the ratio of dispersive power of two prisms in an achromatic combination, follow these steps:

### Step 1: Understand the Definitions
- **Mean Deviation**: The mean deviation for a prism is the average deviation of light passing through it, typically calculated for the yellow color (mean color).
- **Dispersive Power**: It is defined as the ratio of the difference in deviation of violet and red light to the deviation of the mean color (yellow). Mathematically, it can be expressed as:

\[ \text{Dispersive Power} = \frac{D_v - D_r}{D_m} \]

where \(D_v\) is the deviation for violet, \(D_r\) is the deviation for red, and \(D_m\) is the deviation for the mean color.

### Step 2: Set Up the Ratios
- Given that the ratio of the mean deviations produced by the two prisms is \(2:3\), we can denote the mean deviations of the two prisms as:

\[ D_{m1} : D_{m2} = 2 : 3 \]

### Step 3: Relate Dispersive Power to Mean Deviation
- Since the dispersive power is inversely proportional to the mean deviation for the mean color, we can express the relationship as follows:

\[ \text{Dispersive Power} \propto \frac{1}{D_m} \]

### Step 4: Calculate the Ratio of Dispersive Powers
- Let the dispersive powers of the two prisms be \(D_{p1}\) and \(D_{p2}\). Since the dispersive power is inversely proportional to the mean deviation, we can write:

\[ D_{p1} : D_{p2} = \frac{1}{D_{m1}} : \frac{1}{D_{m2}} = D_{m2} : D_{m1} \]

- Substituting the values from the mean deviation ratio:

\[ D_{p1} : D_{p2} = 3 : 2 \]

### Step 5: Conclusion
- Therefore, the ratio of the dispersive powers of the two prisms is \(3:2\).

### Final Answer:
The ratio of the dispersive power of the two prisms is \(3:2\).

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