The dispersive power (w) of the material of a prism is (where symbols have their usual meanings)

Options

A. `omega = (delta _V +delta_ R) / 2`
B. `omega = (delta _V +delta_ R )/ [(delta _V +delta_ R )/ 2] `
C. `omega =(delta_R *delta_V )/[(delta _V +delta_ R) / 2]`
D. `omega= (delta_V - delta_R) / [(delta _V +delta_ R) / 2]`

Answer

To find the dispersive power (w) of the material of a prism, we can follow these steps:

### Step 1: Understand the Definition
Dispersive power (w) is defined as the ability of a transparent medium to disperse light. It is mathematically expressed as the ratio of angular dispersion to the mean deviation.

### Step 2: Write Down the Formula
The formula for dispersive power (w) is given by:

\[ w = \frac{\text{Angular Dispersion}}{\text{Mean Deviation}} \]

### Step 3: Define Angular Dispersion
Angular dispersion is the difference in deviation between the violet and red colors when light passes through the prism. It can be expressed as:

\[ \text{Angular Dispersion} = \delta_v - \delta_r \]

where \(\delta_v\) is the deviation for violet light and \(\delta_r\) is the deviation for red light.

### Step 4: Define Mean Deviation
The mean deviation is the average of the deviations for the two colors. It can be expressed as:

\[ \text{Mean Deviation} = \frac{\delta_v + \delta_r}{2} \]

### Step 5: Substitute into the Formula
Now, substituting the expressions for angular dispersion and mean deviation into the formula for dispersive power:

\[ w = \frac{\delta_v - \delta_r}{\frac{\delta_v + \delta_r}{2}} \]

### Step 6: Simplify the Expression
To simplify the expression, we can multiply both the numerator and the denominator by 2:

\[ w = \frac{2(\delta_v - \delta_r)}{\delta_v + \delta_r} \]

### Step 7: Final Expression
This gives us the final expression for the dispersive power of the material of the prism:

\[ w = \frac{2(\delta_v - \delta_r)}{\delta_v + \delta_r} \]