The speed of electromagnetic wave in vacuum

Options

A. Increases as we move from `gamma`-rays to radio waves
B. Decreases as move from `gamma`-rays to radio waves
C. Is same for all of them
D. None of these

Explanation

To find the speed of electromagnetic waves in a vacuum, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Formula**: The speed of electromagnetic waves in a vacuum is given by the formula:
\[ c = \frac{1}{\sqrt{\epsilon_0 \mu_0}} \]
where \(c\) is the speed of light, \(\epsilon_0\) is the permittivity of free space, and \(\mu_0\) is the permeability of free space.

2. **Values of Constants**: The values of the constants are:
- \(\epsilon_0 \approx 8.854 \times 10^{-12} \, \text{F/m}\) (farads per meter)
- \(\mu_0 \approx 4\pi \times 10^{-7} \, \text{H/m}\) (henries per meter)

3. **Calculating the Speed**: Plugging in the values of \(\epsilon_0\) and \(\mu_0\) into the formula:
\[ c = \frac{1}{\sqrt{(8.854 \times 10^{-12}) \cdot (4\pi \times 10^{-7})}} \]

4. **Simplifying the Calculation**: First, calculate the product:
\[ \epsilon_0 \mu_0 \approx 8.854 \times 10^{-12} \times 4\pi \times 10^{-7} \approx 1.112 \times 10^{-18} \]
Then, take the square root:
\[ \sqrt{\epsilon_0 \mu_0} \approx \sqrt{1.112 \times 10^{-18}} \approx 1.054 \times 10^{-9} \]

5. **Final Calculation of Speed**: Now, calculate \(c\):
\[ c = \frac{1}{1.054 \times 10^{-9}} \approx 2.84 \times 10^{8} \, \text{m/s} \]
However, this is an approximation. The accepted value of \(c\) is:
\[ c \approx 3 \times 10^8 \, \text{m/s} \]

6. **Conclusion**: The speed of electromagnetic waves in a vacuum is approximately:
\[ c \approx 3 \times 10^8 \, \text{m/s} \]