A glass slab has a critical angle of `30^(@)` when placed in air. What will be the critical angle when it is placed in liquid of refractive index `6//5`?

Options

A. `45^(@)`
B. `37^(@)`
C. `53^(@)`
D. `60^(@)`

Explanation

To find the critical angle of a glass slab when placed in a liquid of refractive index \( \frac{6}{5} \), we can follow these steps:

### Step 1: Understand the concept of critical angle
The critical angle \( C \) is defined as the angle of incidence above which total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index. The critical angle can be calculated using the formula:
\[ \sin C = \frac{1}{n} \]
where \( n \) is the refractive index of the second medium (in this case, air or liquid).

### Step 2: Calculate the refractive index of the glass
Given that the critical angle when the glass slab is placed in air is \( 30^\circ \), we can use the formula:
\[ \sin(30^\circ) = \frac{1}{n_{\text{glass}}} \]
Since \( \sin(30^\circ) = \frac{1}{2} \), we can write:
\[ \frac{1}{2} = \frac{1}{n_{\text{glass}}} \]
This implies:
\[ n_{\text{glass}} = 2 \]

### Step 3: Calculate the critical angle in the liquid
Now, we need to find the critical angle when the glass slab is placed in a liquid with a refractive index \( n_{\text{liquid}} = \frac{6}{5} \).

Using the formula for critical angle:
\[ \sin C = \frac{n_{\text{glass}}}{n_{\text{liquid}}} \]
Substituting the known values:
\[ \sin C = \frac{2}{\frac{6}{5}} = 2 \times \frac{5}{6} = \frac{10}{6} = \frac{5}{3} \]
However, since \( \sin C \) cannot exceed 1, we need to check our calculations.

Instead, we should use the relationship:
\[ \sin C = \frac{n_{\text{liquid}}}{n_{\text{glass}}} \]
Thus:
\[ \sin C = \frac{\frac{6}{5}}{2} = \frac{6}{10} = \frac{3}{5} \]

### Step 4: Calculate the angle \( C \)
To find the angle \( C \), we take the inverse sine:
\[ C = \sin^{-1}\left(\frac{3}{5}\right) \]
Using a calculator or trigonometric tables, we find:
\[ C \approx 36.87^\circ \approx 37^\circ \]

### Final Answer
The critical angle when the glass slab is placed in a liquid of refractive index \( \frac{6}{5} \) is approximately \( 37^\circ \).

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