The speed of electromagnetic waves in free space is

Options

A. `mu_0 epsi_0`
B. `(1)/(mu_0 epsi_0)`
C. `sqrt( mu_0 epsi_0)`
D. `(1)/( sqrt((mu_0 epsi_0))`

Answer

To find the speed of electromagnetic waves in free space, we can follow these steps:

### Step 1: Understand the nature of electromagnetic waves
Electromagnetic waves consist of oscillating electric (E) and magnetic (B) fields that are perpendicular to each other and to the direction of wave propagation.

### Step 2: Write the equations for electric and magnetic fields
The electric field can be represented as:
\[ E = E_0 \sin(kx - \omega t) \]
And the magnetic field can be represented as:
\[ B = B_0 \sin(kx - \omega t) \]
where \( E_0 \) and \( B_0 \) are the amplitudes of the electric and magnetic fields, respectively.

### Step 3: Calculate energy density
The energy density \( u \) in an electromagnetic wave can be expressed in terms of the electric and magnetic fields:
- In terms of electric field:
\[ u_E = \frac{1}{2} \epsilon_0 E_0^2 \]
- In terms of magnetic field:
\[ u_B = \frac{1}{2} \frac{B_0^2}{\mu_0} \]

### Step 4: Equate the energy densities
Since both expressions represent the same energy density, we can set them equal to each other:
\[ \frac{1}{2} \epsilon_0 E_0^2 = \frac{1}{2} \frac{B_0^2}{\mu_0} \]

### Step 5: Simplify the equation
Canceling the \( \frac{1}{2} \) from both sides gives:
\[ \epsilon_0 E_0^2 = \frac{B_0^2}{\mu_0} \]

### Step 6: Relate E and B fields
We can express the ratio of the electric field amplitude to the magnetic field amplitude:
\[ \frac{E_0}{B_0} = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

### Step 7: Define the speed of electromagnetic waves
The speed of electromagnetic waves \( c \) in free space is defined as:
\[ c = \frac{E_0}{B_0} \]
Thus, substituting the expression we found:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

### Conclusion
The speed of electromagnetic waves in free space is given by:
\[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]