For small angled prisms, (whose refracting angle (A) is less than 5"), the correct relation between refractive index (`mu`) of material of prism and angle of deviation is

Options

A. `delta = A(mu - 1)`
B. `delta =mu4`
C. `delta = A(mu + 1)`
D. ` delta= A(mu^2 -1)`

Answer

To solve the problem, we need to derive the relationship between the refractive index (μ) of a small angled prism and the angle of deviation (δ) when the refracting angle (A) is less than 5 degrees.

### Step-by-Step Solution:

1. **Understanding the Minimum Deviation Condition**:
For small angled prisms, we consider the condition of minimum deviation. At this point, the angle of incidence (i) and the angle of emergence (e) are equal, and the light ray passes symmetrically through the prism.

2. **Using the Refractive Index Formula**:
The refractive index (μ) of the prism in terms of the angle of deviation (δ) and the refracting angle (A) is given by the formula:
\[ \mu = \frac{\sin\left(\frac{A + \delta}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Applying Small Angle Approximation**:
Since A is small (less than 5 degrees), we can use the small angle approximation, where \(\sin x \approx x\) for small x. Thus, we can rewrite the equation:
\[ \mu \approx \frac{\frac{A + \delta}{2}}{\frac{A}{2}} = \frac{A + \delta}{A} \]

4. **Simplifying the Expression**:
This simplifies to:
\[ \mu = 1 + \frac{\delta}{A} \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ \delta = (\mu - 1) A \]

6. **Final Relation**:
Thus, the final relation between the angle of deviation (δ) and the refractive index (μ) for small angled prisms is:
\[ \delta = (\mu - 1) A \]

### Conclusion:
The correct relation between the refractive index (μ) of the material of the prism and the angle of deviation (δ) is:
\[ \delta = (\mu - 1) A \]