The Brewster angle for the glass air interface is `54.74^(@)` if a ray of light going from air to glass strikes at an angle of incidence `45^(@)` then the angle of refraction is

Options

A. (a)`60^(@)`
B. (b)`30^(@)`
C. (c)`25^(@)`
D. (d)`54.74^(@)`

Answer

To find the angle of refraction when a ray of light strikes the glass-air interface at an angle of incidence of \(45^\circ\), we can use Snell's Law and the information about the Brewster angle.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Brewster angle (\( \theta_B \)) = \(54.74^\circ\)
- Angle of incidence (\( I \)) = \(45^\circ\)

2. **Calculate the Refractive Index:**
- The refractive index (\( \mu \)) can be calculated using the Brewster angle:
\[ \mu = \tan(\theta_B) \]
- Substitute \( \theta_B \):
\[ \mu = \tan(54.74^\circ) \]
- Using a calculator:
\[ \mu \approx 1.5 \]

3. **Apply Snell's Law:**
- Snell's Law states:
\[ \mu_1 \sin(I) = \mu_2 \sin(R) \]
- Here, \( \mu_1 = 1 \) (for air), \( \mu_2 = 1.5 \) (for glass), \( I = 45^\circ \), and \( R \) is the angle of refraction we need to find.
- Substitute the known values:
\[ 1 \cdot \sin(45^\circ) = 1.5 \cdot \sin(R) \]

4. **Calculate \( \sin(45^\circ) \):**
- We know that:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \approx 0.707 \]
- Substitute this into the equation:
\[ 0.707 = 1.5 \cdot \sin(R) \]

5. **Solve for \( \sin(R) \):**
- Rearranging gives:
\[ \sin(R) = \frac{0.707}{1.5} \approx 0.471 \]

6. **Find the Angle of Refraction \( R \):**
- Now, we need to find \( R \) such that:
\[ R = \arcsin(0.471) \]
- Using a calculator:
\[ R \approx 28.1^\circ \]

### Final Answer:
The angle of refraction \( R \) is approximately \( 28.1^\circ \).