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. `60^(@)`
B. `30^(@)`
C. `25^(@)`
D. `54.74^(@)`

Explanation

To solve the problem, we need 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 will use Snell's law and the given Brewster angle to find the refractive index of glass.

### Step-by-Step Solution:

1. **Understand the Brewster Angle**:
The Brewster angle (\(I_p\)) is given as \(54.74^\circ\). This angle is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection.

2. **Calculate the Refractive Index**:
According to Brewster's law, the refractive index (\(\mu\)) can be calculated using the formula:
\[ \mu = \tan(I_p) \]
Substituting the given Brewster angle:
\[ \mu = \tan(54.74^\circ) \]
From trigonometric tables or a calculator, we find:
\[ \tan(54.74^\circ) = \sqrt{2} \]
Therefore, the refractive index of glass is:
\[ \mu = \sqrt{2} \]

3. **Apply Snell's Law**:
Snell's law states:
\[ \mu_1 \sin(I) = \mu_2 \sin(R) \]
Where:
- \(\mu_1\) is the refractive index of air (approximately 1),
- \(I\) is the angle of incidence (\(45^\circ\)),
- \(\mu_2\) is the refractive index of glass (\(\sqrt{2}\)),
- \(R\) is the angle of refraction.

Substituting the known values:
\[ 1 \cdot \sin(45^\circ) = \sqrt{2} \cdot \sin(R) \]

4. **Calculate \(\sin(45^\circ)\)**:
We know:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]
Thus, substituting this into the equation gives:
\[ \frac{1}{\sqrt{2}} = \sqrt{2} \cdot \sin(R) \]

5. **Rearranging the Equation**:
To isolate \(\sin(R)\):
\[ \sin(R) = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2} \]

6. **Finding the Angle of Refraction**:
Now we need to find \(R\) using the inverse sine function:
\[ R = \sin^{-1}\left(\frac{1}{2}\right) \]
From trigonometric values, we know:
\[ R = 30^\circ \]

### Final Answer:
The angle of refraction \(R\) is \(30^\circ\).

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