A ray of light is incident on the surface of seperation of a medium at an angle `45^(@)` and is refracted in the medium at an angle `30^(@)`. What will be the velocity of light in the medium?

Options

A. `1.96xx10^(8)` m/s
B. `2.12xx10^(8)` m/s
C. `2.65xx10^(8)` m/s
D. `1.24xx10^(8)` m/s

Answer

To solve the problem step by step, we will use Snell's Law and the relationship between the speed of light in different media.

### Step 1: Identify the given values
- Angle of incidence (I) = 45°
- Angle of refraction (R) = 30°
- Speed of light in vacuum (C) = 3 × 10^8 m/s

### Step 2: Use Snell's Law
Snell's Law states that:
\[ \mu = \frac{\sin I}{\sin R} \]
where \( \mu \) is the refractive index of the medium.

### Step 3: Calculate the refractive index
Substituting the values into Snell's Law:
\[ \mu = \frac{\sin(45°)}{\sin(30°)} \]

Using the known values:
- \( \sin(45°) = \frac{1}{\sqrt{2}} \)
- \( \sin(30°) = \frac{1}{2} \)

Now substituting these values:
\[ \mu = \frac{\frac{1}{\sqrt{2}}}{\frac{1}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Step 4: Relate the refractive index to the speed of light
The refractive index is also related to the speed of light in vacuum (C) and the speed of light in the medium (v) by the formula:
\[ \mu = \frac{C}{v} \]

### Step 5: Rearranging to find the speed of light in the medium
Rearranging the equation gives:
\[ v = \frac{C}{\mu} \]

### Step 6: Substitute the values
Now substituting the values of C and \( \mu \):
\[ v = \frac{3 \times 10^8 \text{ m/s}}{\sqrt{2}} \]

### Step 7: Calculate the value of v
Calculating \( v \):
\[ v = 3 \times 10^8 \times \frac{1}{\sqrt{2}} \]
\[ v \approx 3 \times 10^8 \times 0.7071 \]
\[ v \approx 2.1213 \times 10^8 \text{ m/s} \]

### Final Answer
Thus, the velocity of light in the medium is approximately:
\[ v \approx 2.12 \times 10^8 \text{ m/s} \]