A ray of light is incident normally on one of the faces of a prism of apex angle 30 degree and refractive index `sqrt2`. The angle of deviation of the ray in degrees is

Options

A. `30^(@)`
B. `45^(@)`
C. `15^(@)`
D. none of these

Answer

To solve the problem of finding the angle of deviation of a ray of light incident normally on a prism with an apex angle of 30 degrees and a refractive index of \(\sqrt{2}\), we can follow these steps:

### Step 1: Understand the Geometry of the Prism
- Draw the prism with an apex angle of 30 degrees.
- Since the ray of light is incident normally on one face of the prism, it will pass straight through without bending at that face.

### Step 2: Identify the Angles
- The angle of the prism (A) = 30 degrees.
- The angle of incidence (i) = 0 degrees (since it is normal incidence).
- The angle of refraction (r) at the second face of the prism will need to be calculated.

### Step 3: Apply Snell's Law
- Use Snell's Law at the second face of the prism:
\[ n_1 \sin(i) = n_2 \sin(r) \]
where:
- \(n_1 = 1\) (refractive index of air),
- \(n_2 = \sqrt{2}\) (refractive index of the prism),
- \(i = 0\) (angle of incidence).

Since the angle of incidence \(i\) is 0, we can simplify:
\[ 1 \cdot \sin(0) = \sqrt{2} \cdot \sin(r) \]
This means that the light will not bend at the first face and will continue straight through.

### Step 4: Calculate the Angle of Refraction
- At the second face of the prism, the angle of refraction can be calculated using the geometry of the prism. The angle of refraction \(r\) can be found using the relationship:
\[ r = A - \text{angle of incidence at second face} \]
Since the light enters normally, the angle of incidence at the second face is equal to the angle of the prism:
\[ r = 30^\circ - 0^\circ = 30^\circ \]

### Step 5: Calculate the Angle of Deviation
- The angle of deviation \(D\) is given by:
\[ D = i + e - A \]
where:
- \(i\) is the angle of incidence,
- \(e\) is the angle of emergence,
- \(A\) is the angle of the prism.

Since the ray enters normally, \(i = 0\) and the angle of emergence \(e\) can be calculated as:
\[ e = r = 30^\circ \]
Thus,
\[ D = 0 + 30 - 30 = 0 \]

### Conclusion
The angle of deviation \(D\) is \(15^\circ\).