# Electromagnetic waves

# Speed of propagation of energy

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A detailed solution is then proposed.

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An electric field which has the structure of a harmonic progressive plane wave, of amplitude , of angular frequency , of the wave vector , propagating along Oz and polarized along Ox.

## Question

Write the expression of the electric field .

### Hint

### Solution

In complex notation :

## Question

Calculating the mean value of the volume density of electromagnetic energy at a point in space.

### Hint

### Solution

The magnetic field is calculated using the relationship of structure :

So :

The average value of the volume density of electromagnetic energy is then :

Either, with :

## Question

Calculate the average value of Poynting vector.

### Hint

### Solution

The average value of the Poynting vector is :

## Question

Calculate the average value of energy on a volume of surface and thickness .

### Hint

### Solution

The average value of the energy in the volume is :

## Question

Calculate the flux of energy through the surface during a time interval .

### Hint

### Solution

The flux of energy through the surface during a time interval is connected to the flux of the Poynting vector, which is a power :

## Question

Derive the value of the speed of propagation of electromagnetic energy. Comment.

### Hint

### Solution

If is the velocity of propagation of energy, then :

Equating the two energies calculated to the previous questions :

It remains :

Thus, in the vacuum (non-dispersive medium), the velocity of propagation is that of the propagation of the wave, or .

Phase and group velocities (often corresponding to the speed of propagation of energy) are equal.