The critical angles of three transpararent media K, L & M are `30^(@), 60^(@)" and "45^(@)` respectively. If `K_(P), L_(P)" and "M_(P)` are their polarising angles respectively, arrange them in increasing order

Options

A. `K_(P), L_(P), M_(P)`
B. `M_(P), L_(P), K_(P)`
C. `L_(P), M_(P), K_(P)`
D. `K_(P), M_(P), L_(P)`

Answer

To solve the problem of arranging the polarizing angles of the three transparent media K, L, and M in increasing order, we will follow these steps:

### Step 1: Understand the relationship between critical angle and polarizing angle
The polarizing angle \( P \) is related to the critical angle \( C \) by the formula:
\[ \tan(P) = \frac{1}{\sin(C)} \]

### Step 2: Calculate the polarizing angle for medium K
Given that the critical angle for medium K (\( C_K \)) is \( 30^\circ \):
\[ \tan(P_K) = \frac{1}{\sin(30^\circ)} \]
Since \( \sin(30^\circ) = \frac{1}{2} \):
\[ \tan(P_K) = \frac{1}{\frac{1}{2}} = 2 \]
Thus,
\[ P_K = \tan^{-1}(2) \approx 63.43^\circ \]

### Step 3: Calculate the polarizing angle for medium L
Given that the critical angle for medium L (\( C_L \)) is \( 60^\circ \):
\[ \tan(P_L) = \frac{1}{\sin(60^\circ)} \]
Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \):
\[ \tan(P_L) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]
Thus,
\[ P_L = \tan^{-1}\left(\frac{2}{\sqrt{3}}\right) \approx 49.1^\circ \]

### Step 4: Calculate the polarizing angle for medium M
Given that the critical angle for medium M (\( C_M \)) is \( 45^\circ \):
\[ \tan(P_M) = \frac{1}{\sin(45^\circ)} \]
Since \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \):
\[ \tan(P_M) = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2} \]
Thus,
\[ P_M = \tan^{-1}(\sqrt{2}) \approx 54.73^\circ \]

### Step 5: Arrange the polarizing angles in increasing order
Now we have:
- \( P_K \approx 63.43^\circ \)
- \( P_L \approx 49.1^\circ \)
- \( P_M \approx 54.73^\circ \)

Arranging these values in increasing order:
1. \( P_L \approx 49.1^\circ \)
2. \( P_M \approx 54.73^\circ \)
3. \( P_K \approx 63.43^\circ \)

### Final Answer
The increasing order of the polarizing angles is:
\[ P_L < P_M < P_K \]