Suppose `theta ` is the polarizing angle for a transparent medium and the speed of light in that medium is v. (Then according to Brewster law )

Options

A. `theta = cot^(-1) (v//c)`
B. `theta =cos^(-1)(v//c)`
C. `theta = sin^(-1) (v//c)`
D. `theta = "cosec"^(-1)(v//c)`

Answer

To solve the problem step by step, we will use Brewster's law and the relationship between the speed of light in different media.

### Step 1: Understand Brewster's Law
Brewster's law states that the polarizing angle (θ) for a transparent medium is given by the formula:
\[ \theta = \tan^{-1}(\mu) \]
where \( \mu \) is the refractive index of the medium.

### Step 2: Relate Refractive Index to Speed of Light
The refractive index \( \mu \) of a medium can be defined as the ratio of the speed of light in vacuum (c) to the speed of light in the medium (v):
\[ \mu = \frac{c}{v} \]

### Step 3: Substitute the Refractive Index into Brewster's Law
Now, substituting the expression for \( \mu \) into Brewster's law, we get:
\[ \theta = \tan^{-1}\left(\frac{c}{v}\right) \]

### Step 4: Use the Identity for Tangent
We know that:
\[ \tan(\theta) = \frac{1}{\cot(\theta)} \]
Thus, we can express \( \cot(\theta) \) in terms of \( v \) and \( c \):
\[ \tan(\theta) = \frac{c}{v} \implies \cot(\theta) = \frac{v}{c} \]

### Step 5: Final Expression
From the above steps, we can conclude that:
\[ \theta = \tan^{-1}\left(\frac{c}{v}\right) \]
and equivalently,
\[ \cot(\theta) = \frac{v}{c} \]

### Conclusion
The polarizing angle \( \theta \) can be expressed in terms of the speed of light in the medium and the speed of light in vacuum.