A ray of light incidents on a refracting surface at `30^(@)` with the surface . If the angle of refraction is `45^(@)` refractive index of the medium is

Options

A. `sqrt(3)`
B. `(sqrt(3))/2`
C. `sqrt(3/2)`
D. `2sqrt(3)`

Answer

To find the refractive index of the medium, we can follow these steps:

### Step 1: Understand the angles involved
The angle of incidence (i) is the angle between the incident ray and the normal to the surface. The angle of refraction (r) is the angle between the refracted ray and the normal.

Given:
- The angle of incidence with the surface = 30°
- The angle of refraction = 45°

### Step 2: Calculate the angle of incidence
Since the angle of incidence is given with respect to the surface, we can find the angle of incidence with respect to the normal:
\[ i = 90° - 30° = 60° \]

### Step 3: Use Snell's Law
Snell's Law states that:
\[ \mu = \frac{\sin i}{\sin r} \]
where:
- \(\mu\) is the refractive index,
- \(i\) is the angle of incidence,
- \(r\) is the angle of refraction.

### Step 4: Substitute the known values
Now, substituting the values of \(i\) and \(r\):
- \(i = 60°\)
- \(r = 45°\)

We have:
\[ \mu = \frac{\sin 60°}{\sin 45°} \]

### Step 5: Calculate the sine values
Using the known values:
- \(\sin 60° = \frac{\sqrt{3}}{2}\)
- \(\sin 45° = \frac{1}{\sqrt{2}}\)

### Step 6: Substitute the sine values into the equation
Now, substituting these values into the equation:
\[ \mu = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{\sqrt{2}}} \]

### Step 7: Simplify the expression
To simplify:
\[ \mu = \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{1} = \frac{\sqrt{3} \cdot \sqrt{2}}{2} = \frac{\sqrt{6}}{2} \]

### Step 8: Final answer
Thus, the refractive index of the medium is:
\[ \mu = \frac{\sqrt{6}}{2} \]