Electromagnetic travel in a medium at a speed of `2.0xx10^8ms^-1`. The relative permitivity of the medium is 2.25. Find the relative permeability of the medium.

Answer

To solve the problem of finding the relative permeability of the medium given the speed of electromagnetic waves and the relative permittivity, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- Speed of electromagnetic wave in the medium, \( V = 2.0 \times 10^8 \, \text{m/s} \)
- Relative permittivity of the medium, \( \epsilon_r = 2.25 \)
- Speed of light in vacuum, \( C = 3.0 \times 10^8 \, \text{m/s} \)

2. **Use the Formula for Wave Speed**:
The speed of electromagnetic waves in a medium is given by the formula:
\[ V = \frac{C}{\sqrt{\mu_r \epsilon_r}} \]
where \( \mu_r \) is the relative permeability and \( \epsilon_r \) is the relative permittivity.

3. **Rearranging the Formula**:
To find the relative permeability \( \mu_r \), we can rearrange the formula:
\[ V^2 = \frac{C^2}{\mu_r \epsilon_r} \]
This can be rearranged to express \( \mu_r \):
\[ \mu_r = \frac{C^2}{V^2 \epsilon_r} \]

4. **Substituting the Known Values**:
Now, substitute the known values into the equation:
\[ \mu_r = \frac{(3.0 \times 10^8)^2}{(2.0 \times 10^8)^2 \times 2.25} \]

5. **Calculating the Squares**:
Calculate \( (3.0 \times 10^8)^2 \) and \( (2.0 \times 10^8)^2 \):
\[ (3.0 \times 10^8)^2 = 9.0 \times 10^{16} \]
\[ (2.0 \times 10^8)^2 = 4.0 \times 10^{16} \]

6. **Substituting Back**:
Substitute these values back into the equation for \( \mu_r \):
\[ \mu_r = \frac{9.0 \times 10^{16}}{4.0 \times 10^{16} \times 2.25} \]

7. **Calculating the Denominator**:
Calculate the denominator:
\[ 4.0 \times 10^{16} \times 2.25 = 9.0 \times 10^{16} \]

8. **Final Calculation**:
Now substitute this back into the equation:
\[ \mu_r = \frac{9.0 \times 10^{16}}{9.0 \times 10^{16}} = 1 \]

9. **Conclusion**:
Therefore, the relative permeability of the medium is:
\[ \mu_r = 1 \]