A ray of light passes through an equilateral prism `(mu=1.5)`. The angle of minimum deviation is (Given `sin 48^(@)36'=0.75`)

Options

A. `45^(@)`
B. `37^(@)12'`
C. `20^(@)`
D. `30^(@)`

Explanation

To solve the problem of finding the angle of minimum deviation for a ray of light passing through an equilateral prism with a refractive index (μ) of 1.5, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Equilateral Prism**:
- An equilateral prism has all its angles equal to 60 degrees. Therefore, the angle of the prism (A) is 60°.

2. **Using the Formula for Minimum Deviation**:
- The formula relating the refractive index (μ), the angle of the prism (A), and the angle of minimum deviation (δm) is:
\[ \mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

3. **Substituting Known Values**:
- Here, we know:
- μ = 1.5
- A = 60°
- Thus, we can substitute these values into the formula:
\[ 1.5 = \frac{\sin\left(\frac{60° + \delta_m}{2}\right)}{\sin\left(\frac{60°}{2}\right)} \]
- Since \(\frac{60°}{2} = 30°\), we have:
\[ 1.5 = \frac{\sin\left(30° + \frac{\delta_m}{2}\right)}{\sin(30°)} \]

4. **Calculating \(\sin(30°)\)**:
- We know that \(\sin(30°) = \frac{1}{2}\). Therefore, the equation becomes:
\[ 1.5 = \frac{\sin\left(30° + \frac{\delta_m}{2}\right)}{\frac{1}{2}} \]
- This simplifies to:
\[ \sin\left(30° + \frac{\delta_m}{2}\right) = 1.5 \times \frac{1}{2} = 0.75 \]

5. **Finding the Angle**:
- We need to find the angle whose sine is 0.75. From the problem statement, we are given that:
\[ \sin(48°36') = 0.75 \]
- Therefore, we can set:
\[ 30° + \frac{\delta_m}{2} = 48°36' \]

6. **Solving for \(\delta_m\)**:
- Rearranging gives:
\[ \frac{\delta_m}{2} = 48°36' - 30° \]
- Converting 30° to minutes:
\[ 30° = 30°00' = 30°00' \text{ (in minutes)} \]
- Thus:
\[ 48°36' - 30°00' = 18°36' \]
- Therefore:
\[ \frac{\delta_m}{2} = 18°36' \]
- Multiplying both sides by 2 gives:
\[ \delta_m = 37°12' \]

### Final Answer:
The angle of minimum deviation (δm) is \(37°12'\).