When light is incident from air to glass at an angle `57^@`, the reflected beam is completely polarised. If the same beam is incident from water to glass, the angle of incidence at which reflected beam is completely polarised will be

Options

A. `theta lt 57^@`
B. `thetagt57^@`
C. `theta=57^@`
D. Can’t be determined

Answer

To solve the problem, we will use Brewster's Law, which states that the angle of incidence at which light is completely polarized upon reflection is known as Brewster's angle (θ_B). This angle can be determined using the formula:

\[ \tan(\theta_B) = \frac{n_2}{n_1} \]

where \( n_1 \) is the refractive index of the first medium (from which the light is coming) and \( n_2 \) is the refractive index of the second medium (into which the light is entering).

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The angle of incidence from air to glass where the reflected beam is completely polarized is given as \( 57^\circ \).
- The refractive index of air (\( n_{air} \)) is approximately \( 1.0 \).
- The refractive index of glass (\( n_{glass} \)) is typically around \( 1.5 \).

2. **Calculate Brewster's Angle for Air to Glass:**
Using Brewster's Law:
\[ \tan(57^\circ) = \frac{n_{glass}}{n_{air}} \]
We can find the refractive index of glass:
\[ n_{glass} = n_{air} \cdot \tan(57^\circ) \]
Since \( n_{air} \approx 1.0 \):
\[ n_{glass} = \tan(57^\circ) \approx 1.5 \]
(This confirms our assumption about the refractive index of glass.)

3. **Consider the Second Case (Water to Glass):**
- The refractive index of water (\( n_{water} \)) is approximately \( 1.33 \).
- We need to find the angle of incidence (\( I_P \)) for the case where light is incident from water to glass.

4. **Apply Brewster's Law for Water to Glass:**
Using Brewster's Law again:
\[ \tan(I_P) = \frac{n_{glass}}{n_{water}} \]
Substituting the known values:
\[ \tan(I_P) = \frac{1.5}{1.33} \]

5. **Calculate \( I_P \):**
\[ I_P = \tan^{-1}\left(\frac{1.5}{1.33}\right) \]
Now, calculate the value:
\[ I_P \approx \tan^{-1}(1.126) \approx 48.37^\circ \]

6. **Conclusion:**
The angle of incidence at which the reflected beam is completely polarized when light is incident from water to glass is approximately \( 48.37^\circ \).