If the polarizing angle of a piece of glass for green light is `54.74^(@)`, then the angle of minimum deviation for an equilateral prism made of same glass is : `["Given" : tan 54.74^(@)=1.414]`

Options

A. `45^(@)`
B. `54.74^(@)`
C. `60^(@)`
D. `30^(@)`

Answer

To solve the problem, we need to find the angle of minimum deviation (Δm) for an equilateral prism made of glass, given that the polarizing angle (θp) for green light is 54.74°.

### Step-by-Step Solution:

1. **Understanding the Polarizing Angle**:
The polarizing angle (θp) is given as 54.74°. This angle is related to the refractive index (μ) of the material by the formula:
\[ \mu = \tan(\theta_p) \]

2. **Calculating the Refractive Index (μ)**:
Using the given value of tan(54.74°) = 1.414, we can find the refractive index:
\[ \mu = \tan(54.74°) = 1.414 \]

3. **Using the Relation for Minimum Deviation**:
For an equilateral prism, the angle of the prism (A) is 60°. The relationship between the refractive index (μ), the angle of the prism (A), and the angle of minimum deviation (Δm) is given by:
\[ \mu = \frac{\sin\left(\frac{A + \Delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \]

4. **Substituting Values**:
Here, A = 60°. Therefore:
\[ \frac{A}{2} = 30° \]
Substituting into the equation:
\[ 1.414 = \frac{\sin\left(30° + \frac{\Delta_m}{2}\right)}{\sin(30°)} \]

5. **Calculating sin(30°)**:
We know that:
\[ \sin(30°) = \frac{1}{2} \]
Thus, the equation becomes:
\[ 1.414 = \frac{\sin\left(30° + \frac{\Delta_m}{2}\right)}{\frac{1}{2}} \]
This simplifies to:
\[ \sin\left(30° + \frac{\Delta_m}{2}\right) = 1.414 \times \frac{1}{2} = 0.707 \]

6. **Finding the Angle**:
The value of sin(45°) = 0.707. Therefore:
\[ 30° + \frac{\Delta_m}{2} = 45° \]

7. **Solving for Δm**:
Rearranging gives:
\[ \frac{\Delta_m}{2} = 45° - 30° = 15° \]
Thus:
\[ \Delta_m = 2 \times 15° = 30° \]

### Final Answer:
The angle of minimum deviation (Δm) for the equilateral prism made of the same glass is **30°**.