If a light ray incidents normally on one of the faces of the prism of refractive index 2 and the emergent ray just grazes the second face of the prism, then the angle of deviation is

Options

A. `0^(@)`
B. `30^(@)`
C. `45^(@)`
D. `60^(@)`

Answer

To solve the problem step-by-step, we will analyze the situation of light passing through a prism with a refractive index of 2, where the light ray is incident normally on one face and the emergent ray grazes the second face.

### Step 1: Understand the conditions
- The light ray is incident normally on one face of the prism, which means the angle of incidence (i) is 0 degrees.
- The emergent ray grazes the second face of the prism, which means it is at the critical angle (c) at that face.

### Step 2: Determine the critical angle
The critical angle (c) can be calculated using Snell's law:
\[ n_1 \sin(i) = n_2 \sin(r) \]
Where:
- \( n_1 = 1 \) (refractive index of air)
- \( n_2 = 2 \) (refractive index of the prism)
- \( i \) is the angle of incidence at the second face (which is the critical angle here)
- \( r = 90^\circ \) (since the ray grazes the surface)

Using Snell's law:
\[ 1 \cdot \sin(c) = 2 \cdot \sin(90^\circ) \]
\[ \sin(c) = 2 \cdot 1 \]
\[ \sin(c) = \frac{1}{2} \]

From this, we find:
\[ c = \sin^{-1}\left(\frac{1}{2}\right) = 30^\circ \]

### Step 3: Identify angles
- The angle of incidence at the first face (i) = 0 degrees (since it is normal incidence).
- The angle of refraction at the first face (r) can be calculated using Snell's law:
\[ n_1 \sin(i) = n_2 \sin(r) \]
\[ 1 \cdot \sin(0) = 2 \cdot \sin(r) \]
This means \( r = 0^\circ \) (the light continues straight through).

- At the second face, the angle of incidence (i') is equal to the critical angle (c) = 30 degrees.

### Step 4: Calculate the angle of deviation
The formula for the angle of deviation (D) is given by:
\[ D = i + i' - r - r' \]
Where:
- \( i = 0^\circ \) (angle of incidence at the first face)
- \( i' = 90^\circ \) (angle of refraction at the second face, since it grazes)
- \( r = 30^\circ \) (angle of incidence at the second face)
- \( r' = 0^\circ \) (angle of refraction at the first face)

Substituting the values:
\[ D = 0 + 90 - 30 - 0 \]
\[ D = 90 - 30 \]
\[ D = 60^\circ \]

### Conclusion
The angle of deviation is \( 60^\circ \).