A ray of light incident on a transparent sphere at an angle `pi//4` and refracted at an angle r, emerges from the sphere after suffering one internal reflection. The total angle of deviation of the ray is

Options

A. `(3pi)/(2) - 4r`
B. `(pi)/(2) - 4r`
C. `(pi)/(4) - r`
D. `(5pi)/(2) - 4r`

Explanation

To solve the problem, we will analyze the path of the ray of light as it enters and exits the transparent sphere, taking into account the angles of incidence, refraction, and reflection.

### Step 1: Identify the angles
The ray of light is incident on the sphere at an angle of \( \frac{\pi}{4} \). Let the angle of refraction inside the sphere be \( r \).

### Step 2: Apply Snell's Law at the first interface
Using Snell's law at the point of incidence:
\[ n_1 \sin(\theta_1) = n_2 \sin(r) \]
Assuming the sphere is in air, \( n_1 = 1 \) (air) and \( n_2 \) is the refractive index of the sphere. Thus:
\[ \sin\left(\frac{\pi}{4}\right) = n_2 \sin(r) \]
This gives us:
\[ \frac{1}{\sqrt{2}} = n_2 \sin(r) \quad \text{(1)} \]

### Step 3: Internal Reflection
After refraction, the ray travels inside the sphere and strikes the opposite side. The angle of incidence at this point is \( r \). According to the law of reflection, the angle of reflection will also be \( r \).

### Step 4: Calculate the angle of emergence
The ray will then refract out of the sphere at the same angle \( r \) (due to symmetry). Thus, the angle of emergence will also be \( \frac{\pi}{4} \).

### Step 5: Calculate the total angle of deviation
The total angle of deviation \( D \) can be calculated by considering the angles involved:
1. The ray wanted to go straight but deviated at the first interface, which gives a deviation of:
\[ D_1 = \frac{\pi}{4} - r \]
2. After internal reflection, the ray again deviates when it exits the sphere:
\[ D_2 = \pi - 2r \]
3. Finally, the ray deviates again when it exits:
\[ D_3 = \frac{\pi}{4} - r \]

### Step 6: Combine the deviations
The total deviation \( D \) is the sum of all deviations:
\[ D = D_1 + D_2 + D_3 = \left(\frac{\pi}{4} - r\right) + \left(\pi - 2r\right) + \left(\frac{\pi}{4} - r\right) \]
Combining these:
\[ D = \frac{\pi}{4} + \frac{\pi}{4} + \pi - 4r = \frac{3\pi}{2} - 4r \]

### Final Answer
The total angle of deviation of the ray is:
\[ D = \frac{3\pi}{2} - 4r \]