The velocity of electromagnetic radiation in a medium of permittivity `epsilon_(0)` and permeability `mu_(0)` is given by:

Options

A. `sqrt(epsilon_(0)/(mu_(0))`
B. `sqrt(mu_(0)epsilon_(0))`
C. `(1)/sqrt(mu_(0)epsilon_(0))`
D. `sqrt(mu_(0)/(epsilon_(0))`

Answer

To find the expression for the velocity of electromagnetic radiation in a medium characterized by permittivity \( \epsilon_0 \) and permeability \( \mu_0 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship**: The velocity \( v \) of electromagnetic waves in a medium is related to its permittivity \( \epsilon \) and permeability \( \mu \) by the formula:
\[ v = \frac{1}{\sqrt{\mu \epsilon}} \]

2. **Substitute the Values**: For free space, we have:
- Permittivity \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \) (Farads per meter)
- Permeability \( \mu_0 = 4\pi \times 10^{-7} \, \text{H/m} \) (Henries per meter)

3. **Calculate \( \mu_0 \epsilon_0 \)**: We need to multiply \( \mu_0 \) and \( \epsilon_0 \):
\[ \mu_0 \epsilon_0 = (4\pi \times 10^{-7}) \times (8.85 \times 10^{-12}) \]

4. **Perform the Multiplication**:
- Calculate \( 4\pi \approx 12.566 \)
- Then multiply:
\[ \mu_0 \epsilon_0 \approx 12.566 \times 10^{-7} \times 8.85 \times 10^{-12} \approx 1.112 \times 10^{-18} \]

5. **Find the Velocity**: Now, substitute back into the velocity formula:
\[ v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = \frac{1}{\sqrt{1.112 \times 10^{-18}}} \]

6. **Calculate \( v \)**:
\[ v \approx 3 \times 10^8 \, \text{m/s} \]

7. **Conclusion**: The speed of electromagnetic radiation in a vacuum (or free space) is approximately \( 3 \times 10^8 \, \text{m/s} \).

### Final Expression:
Thus, the velocity of electromagnetic radiation in a medium characterized by permittivity \( \epsilon_0 \) and permeability \( \mu_0 \) is given by:
\[ v = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \, \text{m/s} \]