The angle of polarisation for any medium is `60^(@)` , what will be critical angle for this

Options

A. `sin^(-1)(sqrt(3))`
B. `tan^(-1)(sqrt(3))`
C. `cos^(-1)(sqrt(3))`
D. `sin^(-1)(1/(sqrt(3)))`

Answer

To find the critical angle for a medium with an angle of polarization of \(60^\circ\), we can follow these steps:

### Step 1: Understand the relationship between angle of polarization and refractive index
The angle of polarization (\(I_P\)) is related to the refractive index (\(\mu\)) of the medium by the formula:
\[ \mu = \tan I_P \]

### Step 2: Substitute the given angle of polarization
Given that the angle of polarization \(I_P = 60^\circ\), we can substitute this value into the equation:
\[ \mu = \tan(60^\circ) \]

### Step 3: Calculate the refractive index
We know that:
\[ \tan(60^\circ) = \sqrt{3} \]
Thus, we can write:
\[ \mu = \sqrt{3} \]

### Step 4: Relate the refractive index to the critical angle
The critical angle (\(C\)) is related to the refractive index by the formula:
\[ \mu = \frac{1}{\sin C} \]

### Step 5: Substitute the value of refractive index
Now, substituting \(\mu = \sqrt{3}\) into the equation:
\[ \sqrt{3} = \frac{1}{\sin C} \]

### Step 6: Solve for \(\sin C\)
Rearranging the equation gives:
\[ \sin C = \frac{1}{\sqrt{3}} \]

### Step 7: Find the critical angle
To find the critical angle \(C\), we take the inverse sine:
\[ C = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

### Conclusion
Thus, the critical angle for the medium is:
\[ C = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

### Final Answer
The critical angle is \(\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)\), which corresponds to option 4.

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