A rectangular block of glass (R.l. = 3/2) is kept in water (R.l. = 4/3). The critical angle for total internal reflection between glass and water is

Options

A. `sin^(-1)(2/3)`
B. `sin^(-1)(1/24)`
C. `tan^(-1)(2/3)`
D. `tan^(-1)(3/2)`

Answer

To find the critical angle for total internal reflection between the glass and water, we can follow these steps:

### Step 1: Identify the refractive indices
- The refractive index of glass (n_glass) is given as \( \frac{3}{2} \).
- The refractive index of water (n_water) is given as \( \frac{4}{3} \).

### Step 2: Understand the condition for total internal reflection
Total internal reflection occurs when light travels from a denser medium (glass) to a rarer medium (water). The critical angle (θ_c) is the angle of incidence in the denser medium at which the angle of refraction in the rarer medium is 90 degrees.

### Step 3: Apply Snell's Law
According to Snell's Law:
\[ n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \]
Where:
- \( n_1 = n_{glass} = \frac{3}{2} \)
- \( n_2 = n_{water} = \frac{4}{3} \)
- \( \theta_1 = \theta_c \) (the critical angle)
- \( \theta_2 = 90^\circ \)

Since \( \sin(90^\circ) = 1 \), we can rewrite Snell's Law as:
\[ \frac{3}{2} \sin(\theta_c) = \frac{4}{3} \cdot 1 \]

### Step 4: Solve for the critical angle
Rearranging the equation gives:
\[ \sin(\theta_c) = \frac{4}{3} \cdot \frac{2}{3} = \frac{8}{9} \]

### Step 5: Calculate the critical angle
To find \( \theta_c \), we take the inverse sine:
\[ \theta_c = \sin^{-1}\left(\frac{8}{9}\right) \]

### Final Answer
Thus, the critical angle for total internal reflection between the glass and water is:
\[ \theta_c = \sin^{-1}\left(\frac{8}{9}\right) \]

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