Brewster angle for air to water transition is (refractive index of water is `4/3`)

Options

A. `sin^-1 3/4`
B. `cos^-1 3/4`
C. `tan^-1 3/5`
D. `cot^-1 3/4`

Answer

To find the Brewster angle for the transition from air to water, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Brewster Angle**: The Brewster angle (θp) is defined as the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. At this angle, the reflected and refracted rays are perpendicular to each other.

2. **Use the Formula**: The relationship between the Brewster angle and the refractive index (μ) of the medium is given by:
\[ \tan(\theta_p) = \mu \]
where μ is the refractive index of the second medium (in this case, water).

3. **Identify the Refractive Index**: According to the problem, the refractive index of water is given as:
\[ \mu = \frac{4}{3} \]

4. **Calculate the Brewster Angle**: Using the formula from step 2, we can express the Brewster angle as:
\[ \theta_p = \tan^{-1}(\mu) = \tan^{-1}\left(\frac{4}{3}\right) \]

5. **Convert to Cotangent**: Since the options provided do not include \(\tan^{-1}\left(\frac{4}{3}\right)\), we can express this in terms of cotangent. We know that:
\[ \tan(\theta) = \frac{1}{\cot(\theta)} \]
Therefore, we can write:
\[ \theta_p = \cot^{-1}\left(\frac{3}{4}\right) \]
This is because if \(\tan(\theta_p) = \frac{4}{3}\), then \(\cot(\theta_p) = \frac{3}{4}\).

6. **Select the Correct Option**: From the options provided, the correct expression for the Brewster angle is:
\[ \theta_p = \cot^{-1}\left(\frac{3}{4}\right) \]
This corresponds to option 4.

### Final Answer:
The Brewster angle for the air to water transition is:
\[ \theta_p = \cot^{-1}\left(\frac{3}{4}\right) \]