A light ray is incident on a transparent sphere of index = `sqrt(2)` , at an angle of incidence =`45^(@)` , What is the deviation of a tiny fraction of the ray, which enters the sphere, undergoes two internal reflections and then refracts out into air?

Options

A. `270^(@)`
B. `240^(@)`
C. `120^(@)`
D. `180^(@)`

Explanation

To solve the problem of finding the deviation of a light ray incident on a transparent sphere with a refractive index of \(\sqrt{2}\) at an angle of incidence of \(45^\circ\), we will follow these steps:

### Step 1: Calculate the angle of refraction at the first interface
Using Snell's law:
\[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]
where:
- \(n_1 = 1\) (refractive index of air)
- \(\theta_1 = 45^\circ\) (angle of incidence)
- \(n_2 = \sqrt{2}\) (refractive index of the sphere)
- \(\theta_2\) is the angle of refraction.

Substituting the values:
\[ 1 \cdot \sin(45^\circ) = \sqrt{2} \cdot \sin(\theta_2) \]
\[ \frac{\sqrt{2}}{2} = \sqrt{2} \cdot \sin(\theta_2) \]
\[ \sin(\theta_2) = \frac{1}{2} \implies \theta_2 = 30^\circ \]

### Step 2: Calculate the deviation at the first interface
The deviation (\(\delta_1\)) at the first interface is given by:
\[ \delta_1 = \theta_1 - \theta_2 = 45^\circ - 30^\circ = 15^\circ \]

### Step 3: Calculate the angles for total internal reflection (TIR)
When the light ray hits the inner surface of the sphere, it will undergo total internal reflection. The angle of incidence at the first internal reflection is \(30^\circ\).

Using the property of reflection:
\[ \text{Angle of reflection} = \text{Angle of incidence} = 30^\circ \]

### Step 4: Calculate the deviation at the first TIR
The deviation (\(\delta_2\)) at the first TIR is given by:
\[ \delta_2 = 180^\circ - 2 \times 30^\circ = 120^\circ \]

### Step 5: Calculate the angles for the second TIR
The light ray will undergo a second total internal reflection with the same angle of incidence \(30^\circ\).

### Step 6: Calculate the deviation at the second TIR
The deviation (\(\delta_3\)) at the second TIR is also:
\[ \delta_3 = 180^\circ - 2 \times 30^\circ = 120^\circ \]

### Step 7: Calculate the angle of refraction at the second interface
After the second TIR, the light ray exits the sphere. The angle of incidence at the second interface is again \(30^\circ\).

Using Snell's law again:
\[ \sqrt{2} \cdot \sin(30^\circ) = 1 \cdot \sin(\theta_4) \]
\[ \sqrt{2} \cdot \frac{1}{2} = \sin(\theta_4) \]
\[ \sin(\theta_4) = \frac{\sqrt{2}}{2} \implies \theta_4 = 45^\circ \]

### Step 8: Calculate the deviation at the second interface
The deviation (\(\delta_4\)) at the second interface is:
\[ \delta_4 = \theta_4 - \theta_2 = 45^\circ - 30^\circ = 15^\circ \]

### Step 9: Calculate the total deviation
The total deviation (\(\delta_{total}\)) is the sum of all deviations:
\[ \delta_{total} = \delta_1 + \delta_2 + \delta_3 + \delta_4 \]
\[ \delta_{total} = 15^\circ + 120^\circ + 120^\circ + 15^\circ = 270^\circ \]

Thus, the total deviation of the tiny fraction of the ray is \(270^\circ\).