Light travelling in air falls at an incidence angle of `2^(@)` on one rafracting surface of a prism of refractive index `1.5` and angle of refraction `4^(@)`. The medium on the other side is water `(n=4//3)` . Find the deviation produced by the prism.

Explanation

To find the deviation produced by the prism, we will follow these steps:

### Step 1: Understand the Problem
We have light traveling from air into a prism and then into water. The refractive indices are given as follows:
- Refractive index of air, \( n_1 = 1 \)
- Refractive index of prism, \( n = 1.5 \)
- Refractive index of water, \( n_2 = \frac{4}{3} \)

The angle of incidence \( i = 2^\circ \) and the angle of refraction \( r_1 = 4^\circ \) when the light enters the prism.

### Step 2: Apply Snell's Law at the First Surface
Using Snell's Law at the first surface (air to prism):
\[ n_1 \sin(i) = n \sin(r_1) \]
Substituting the known values:
\[ 1 \cdot \sin(2^\circ) = 1.5 \cdot \sin(4^\circ) \]

### Step 3: Calculate Angle of Refraction \( r_1 \)
Using the small angle approximation (since angles are small):
\[ \sin(2^\circ) \approx \frac{2\pi}{180} \cdot 2 \approx 0.0349 \quad \text{and} \quad \sin(4^\circ) \approx \frac{2\pi}{180} \cdot 4 \approx 0.06976 \]
Now substituting:
\[ 0.0349 = 1.5 \cdot 0.06976 \]
This confirms that \( r_1 \) is indeed \( 4^\circ \).

### Step 4: Find Angle of Refraction \( r_2 \)
The angle of the prism \( A \) is given as:
\[ A = r_1 + r_2 \]
Given that the angle of the prism is \( 4^\circ \):
\[ 4^\circ = r_1 + r_2 \]
Substituting \( r_1 = 4^\circ \):
\[ r_2 = 4^\circ - r_1 = 4^\circ - \frac{4}{3}^\circ = \frac{8}{3}^\circ \]

### Step 5: Apply Snell's Law at the Second Surface
Using Snell's Law at the second surface (prism to water):
\[ n \sin(r_2) = n_2 \sin(e) \]
Substituting the known values:
\[ 1.5 \cdot \sin\left(\frac{8}{3}^\circ\right) = \frac{4}{3} \cdot \sin(e) \]

### Step 6: Calculate Angle of Emergence \( e \)
Using the small angle approximation again:
\[ \sin\left(\frac{8}{3}^\circ\right) \approx \frac{8\pi}{3 \cdot 180} \approx 0.0461 \]
Now substituting:
\[ 1.5 \cdot 0.0461 = \frac{4}{3} \cdot \sin(e) \]
Solving for \( \sin(e) \):
\[ \sin(e) = \frac{1.5 \cdot 0.0461 \cdot 3}{4} \approx 0.0173 \]
Thus, \( e \approx 3^\circ \).

### Step 7: Calculate the Angle of Deviation \( \delta \)
Using the formula for deviation:
\[ \delta = i + e - A \]
Substituting the values:
\[ \delta = 2^\circ + 3^\circ - 4^\circ = 1^\circ \]

### Final Answer
The deviation produced by the prism is \( \delta = 1^\circ \).