For refraction through a small angled prism, the angle of minimum deviation

Options

A. increases with increase in refractive index of a prism
B. will be `2delta` for a ray of refractive index `2.4` if it is delta for a ray of refractive index `1.2`
C. is directly proportional to the angle of the prism
D. will decrease with increase in refractive index of the prism

Explanation

To solve the problem regarding the angle of minimum deviation for refraction through a small angled prism, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Prism and Refraction**:
- A prism is a transparent optical element that refracts light. The angle of the prism is denoted as 'a', and the angle of minimum deviation is denoted as 'δ_m'.

2. **Using the Formula for Refractive Index**:
- The refractive index (μ) of a prism can be expressed using the formula:
\[ \mu = \frac{\sin\left(\frac{a + \delta_m}{2}\right)}{\sin\left(\frac{a}{2}\right)} \]

3. **Approximation for Small Angles**:
- For small angles, we can use the approximation:
\[ \sin x \approx x \quad \text{(when x is in radians)} \]
- Therefore, for a small angled prism, we can write:
\[ \sin\left(\frac{a}{2}\right) \approx \frac{a}{2} \]

4. **Substituting the Approximation into the Formula**:
- Substituting the approximation into the refractive index formula gives:
\[ \mu = \frac{\sin\left(\frac{a + \delta_m}{2}\right)}{\frac{a}{2}} \]

5. **Rearranging the Equation**:
- Rearranging the equation, we have:
\[ \mu \cdot \frac{a}{2} = \sin\left(\frac{a + \delta_m}{2}\right) \]

6. **Analyzing the Effect of Refractive Index**:
- If the refractive index (μ) increases, the left side of the equation increases. Consequently, this means that the right side must also increase, which implies that the angle of minimum deviation (δ_m) must also increase.

7. **Conclusion**:
- Therefore, we conclude that the angle of minimum deviation increases with an increase in the refractive index of the prism.

### Final Answer:
The angle of minimum deviation increases with an increase in the refractive index of the prism.

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