The refracting angle of a prism is A and the refractive index is cot A/2. The angle of minimum deviation is :

Question

Accepted Answer

The problem involves a prism with a refracting angle $ A $ and a refractive index given by $ \cot \frac{A}{2} $. We are tasked with finding the angle of minimum deviation, $\delta_m$.

To solve this, we use the formula for the angle of minimum deviation in a prism:

$$
\delta_m = 2i - A
$$

where $ i $ is the angle of incidence that results in minimum deviation. For minimum deviation, the angle of incidence $ i $ is equal to the angle of emergence $ e $, and the light ray inside the prism is symmetric with respect to the prism's geometry.

The relationship between the refractive index $ n $, the angle of incidence $ i $, and the angle of refraction $ r $ is given by Snell's law:

$$
n = \frac{\sin i}{\sin r}
$$

For minimum deviation, the angle of refraction $ r $ is $ \frac{A}{2} $. Given that the refractive index $ n = \cot \frac{A}{2} $, we have:

$$
\cot \frac{A}{2} = \frac{\sin i}{\sin \frac{A}{2}}
$$

This implies:

$$
\sin i = \cot \frac{A}{2} \cdot \sin \frac{A}{2} = \cos \frac{A}{2}
$$

Thus, $ i = \frac{\pi}{2} - \frac{A}{2} $.

Substituting $ i = \frac{\pi}{2} - \frac{A}{2} $ into the formula for $\delta_m$:

$$
\delta_m = 2\left(\frac{\pi}{2} - \frac{A}{2}\right) - A
$$

Simplifying this expression:

$$
\delta_m = \pi - A - A = \pi - 2A
$$

Since $\pi$ radians is equivalent to 180°, we have:

$$
\delta_m = 180° - 2A
$$

Therefore, the angle of minimum deviation is $ 180° - 2A $, which corresponds to Option 2.

**Explanation for Incorrect Options:**

1. **Option 1: 180° - A**  
   This option suggests that the deviation is only reduced by the angle $ A $, which does not account for the symmetrical nature of the light path within the prism at minimum deviation.

2. **Option 3: 180° - 3A**  
   This option implies a greater reduction in deviation than is physically possible given the geometry and refractive index of the prism.

3. **Option 4: 180° - 4A**  
   This option suggests an even larger reduction in deviation, which is not supported by the relationship between the refracting angle, refractive index, and the geometry of the prism.

In conclusion, the correct answer is Option 2: $ 180° - 2A $, as it accurately reflects the relationship between the refracting angle, refractive index, and the angle of minimum deviation for the given prism.