Dispersive power of a prism with the increase in prism angle

Options

A. increases
B. decreases
C. remains the same
D. none of these

Answer

To solve the question regarding the dispersive power of a prism with the increase in prism angle, we need to understand the concept of dispersive power and how it relates to the prism angle.

### Step-by-Step Solution:

1. **Understanding Dispersive Power**:
Dispersive power (ω) of a prism is defined as the ratio of the angle of dispersion (Δ) to the mean deviation (Δy) for yellow light. Mathematically, it can be expressed as:
\[ \omega = \frac{\Delta}{\Delta y} \]

2. **Angle of Dispersion**:
The angle of dispersion (Δ) is the difference in the deviation of violet (Δv) and red (Δr) light:
\[ \Delta = \Delta v - \Delta r \]

3. **Deviation of Light**:
The deviation for light passing through a prism can be expressed in terms of the prism angle (A) and the refractive index (μ) of the prism material:
\[ \Delta v = A(\mu_v - 1) \quad \text{and} \quad \Delta r = A(\mu_r - 1) \]

4. **Substituting into the Angle of Dispersion**:
Substituting the expressions for Δv and Δr into the equation for Δ:
\[ \Delta = A(\mu_v - 1) - A(\mu_r - 1) = A(\mu_v - \mu_r) \]

5. **Mean Deviation for Yellow Light**:
The mean deviation for yellow light can also be expressed as:
\[ \Delta y = A(\mu - 1) \]

6. **Calculating Dispersive Power**:
Now substituting Δ and Δy into the formula for dispersive power:
\[ \omega = \frac{A(\mu_v - \mu_r)}{A(\mu - 1)} \]
Here, the angle A cancels out:
\[ \omega = \frac{\mu_v - \mu_r}{\mu - 1} \]

7. **Effect of Increasing Prism Angle**:
Notice that the expression for dispersive power (ω) is independent of the prism angle (A). Therefore, if we increase the prism angle, the dispersive power remains constant.

8. **Conclusion**:
Thus, the dispersive power of the prism does not change with an increase in the prism angle. The correct answer is that the dispersive power will remain the same.

### Final Answer:
The dispersive power of a prism remains the same with the increase in the prism angle.

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