A glass prism of refractive index 1.5 is placed in water of refractive index 1.33. The minimum value of the angle of the prism so that it will not be possible to have any emergent ray is

Options

A. `150^@`
B. `125^@`
C. `165^@`
D. `180^@`

Explanation

To solve the problem of finding the minimum value of the angle of a glass prism placed in water such that no emergent ray is possible, we can follow these steps:

### Step 1: Understand the Problem
We have a glass prism with a refractive index (μ) of 1.5 placed in water with a refractive index of 1.33. We need to determine the minimum angle of the prism (A) such that light entering the prism does not emerge out.

### Step 2: Apply Snell's Law
At the interface between the water and the prism, we can apply Snell's law:
\[ \mu_2 \sin R_2 = \mu_1 \sin R_1 \]
where:
- \( \mu_2 = 1.5 \) (refractive index of glass)
- \( \mu_1 = 1.33 \) (refractive index of water)
- \( R_1 \) is the angle of incidence in the water
- \( R_2 \) is the angle of refraction in the prism

### Step 3: Determine the Condition for No Emergent Ray
For no emergent ray, the angle of refraction \( R_2 \) must be 90 degrees. This is because if \( R_2 \) is greater than 90 degrees, the light will not pass through the prism.

### Step 4: Set Up the Equation
When \( R_2 = 90^\circ \), we can substitute this into Snell's law:
\[ \mu_2 \sin(90^\circ) = \mu_1 \sin R_1 \]
This simplifies to:
\[ \mu_2 = \mu_1 \sin R_1 \]
Substituting the values:
\[ 1.5 = 1.33 \sin R_1 \]

### Step 5: Solve for \( R_1 \)
Rearranging gives:
\[ \sin R_1 = \frac{1.5}{1.33} \]
Calculating this:
\[ \sin R_1 = \frac{1.5}{1.33} \approx 1.126 \]
Since the sine function cannot exceed 1, this indicates that \( R_1 \) must be at its maximum value for total internal reflection to occur.

### Step 6: Find the Minimum Angle of the Prism
From the geometry of the prism, we know:
\[ A = R_1 + R_2 \]
For the minimum angle \( A \), we set \( R_1 = R_2 \) (the condition for minimum angle):
Thus:
\[ A = 2R_2 \]
Since \( R_2 \) is at its critical angle, we can find \( R_2 \) using:
\[ R_2 = \sin^{-1} \left( \frac{1.33}{1.5} \right) \]
Calculating this gives:
\[ R_2 \approx 62.7^\circ \]

### Step 7: Calculate Minimum Angle A
Now substituting back:
\[ A = 2 \times 62.7^\circ \approx 125.4^\circ \]

### Conclusion
The minimum value of the angle of the prism such that it will not be possible to have any emergent ray is approximately:
\[ A \approx 125^\circ \]