What is the velocity of the light in the medium if Sin `theta_i`=0.707 , Sin `theta_r`=0.500 , and the velocity of light in a vacuum is `3.0xx10^8` m/sec ?

Options

A. `2.1xx10^8` m/sec
B. `2.8xx10^8` m/sec
C. `1.4xx10^8` m/sec
D. `4.2xx10^8` m/sec

Explanation

To find the velocity of light in a medium given the values of sin(theta_i) and sin(theta_r), we can use Snell's law, which relates the angles of incidence and refraction to the velocities of light in different media.

### Step-by-Step Solution:

1. **Identify Given Values:**
- \( \sin \theta_i = 0.707 \)
- \( \sin \theta_r = 0.500 \)
- Speed of light in vacuum, \( c = 3.0 \times 10^8 \, \text{m/s} \)

2. **Use Snell's Law:**
Snell's law states that:
\[ \frac{\sin \theta_i}{\sin \theta_r} = \frac{c}{v} \]
where \( v \) is the speed of light in the medium.

3. **Rearranging the Equation:**
We can rearrange the equation to solve for \( v \):
\[ v = c \cdot \frac{\sin \theta_r}{\sin \theta_i} \]

4. **Substituting the Values:**
Substitute the known values into the equation:
\[ v = 3.0 \times 10^8 \cdot \frac{0.500}{0.707} \]

5. **Calculating the Ratio:**
Calculate the ratio:
\[ \frac{0.500}{0.707} \approx 0.707 \]

6. **Final Calculation:**
Now, calculate \( v \):
\[ v \approx 3.0 \times 10^8 \cdot 0.707 \approx 2.121 \times 10^8 \, \text{m/s} \]

7. **Conclusion:**
The velocity of light in the medium is approximately:
\[ v \approx 2.1 \times 10^8 \, \text{m/s} \]