A ray of light is incident on the surface of a glass plate of refractive index `sqrt(3)` at the polarising angle . The angle of incidence and angle of refraction of the ray is

Options

A. `60^(@), 30^(@)`
B. `30^(@), 60^(@)`
C. `sin^(-1)""((1)/(sqrt(3))), 45^(@)`
D. `tan^(-1)""((sqrt(3))/(1)), 30^(@)`

Answer

To solve the problem of finding the angle of incidence and angle of refraction of a ray of light incident on a glass plate at the polarizing angle, we can follow these steps:

### Step 1: Understand the Polarizing Angle
The polarizing angle (also known as Brewster's angle) is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. The relationship between the polarizing angle (p) and the refractive index (μ) of the material is given by:
\[ \tan(p) = \mu \]

### Step 2: Given Data
We are given that the refractive index of the glass plate is:
\[ \mu = \sqrt{3} \]

### Step 3: Calculate the Polarizing Angle
Using the formula for the polarizing angle:
\[ \tan(p) = \sqrt{3} \]
To find the polarizing angle (p), we take the arctangent:
\[ p = \tan^{-1}(\sqrt{3}) \]

### Step 4: Determine the Angle of Incidence
From trigonometric values, we know:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, the polarizing angle is:
\[ p = 60^\circ \]
So, the angle of incidence (i) is:
\[ i = p = 60^\circ \]

### Step 5: Calculate the Angle of Refraction
According to Snell's law, the angle of refraction (r) can be found using:
\[ r = 90^\circ - p \]
Substituting the value of p:
\[ r = 90^\circ - 60^\circ = 30^\circ \]

### Conclusion
Thus, the angle of incidence is \( 60^\circ \) and the angle of refraction is \( 30^\circ \).

### Final Answer:
- Angle of Incidence (i) = \( 60^\circ \)
- Angle of Refraction (r) = \( 30^\circ \)

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