Chapter 2

Maxwell's equations

Electromagnetic energy review

FondamentalDensity of power given by the electromagnetic field to matter

An electromagnetic field will interact with charged particles and give them energy.

In fact, a charge is subject to Lorentz' force from this electromagnetic field, which has a power of :

By writing the number of charge carriers by volume unit, the density of power given by the electromagnetic field to matter is expressed by :

Remark :

The power received by the electromagnetic field from charge carriers is equal to (do the analogy with , the density of power received by a heat-conductor setting from the heat sources, see below).

RappelEnergy conservation equation in conductive phenomena (see lesson about thermal transfers)

Let us consider a volume delimited by a closed surface (steady in the referential of study).

The total internal energy included in the volume at the time t is :

Where is the density of internal energy.

Heat equation (reminders)

The internal energy conservation allows us to write :

The volume is steady, so :

By using the divergence theorem (or Green - Ostrogradsky's law), it comes :

This result is true for any volume , so :

This equation had been proven in the one dimension case.

FondamentalLocal conservation of electromagnetic energy equation

By using the analogy with conservation equations (charge, mass, diffusion, heat) we would like to get an equation of this kind :

Where represents the density of electromagnetic energy (included in the electromagnetic field).

is a vector called the "Poynting vector".

It is supposed to give the direction of the electromagnetic energy exchanges (especially by calculating its flux through a surface).

The following calculation is not on the curriculum :

The product can be expressed as such by using Maxwell - Ampere's equation :

By writing that :

Hence :

Thus :

Then :

Can be written :

  • Electromagnetic density of energy :

  • Poynting vector :

Thus the equation can be written this way :

And thus corresponds to an electromagnetic energy assessment.

AttentionPoynting Energy assessment

  • Density of electromagnetic energy :

  • Poynting vector :

Local conservation of electromagnetic energy :

The integral form of the conservation of electromagnetic energy is :

RemarqueEnergy propagation velocity

By proceeding to an analogy with the charge conservation equation, an energy propagation speed (written ) can be defined with this relation :

(analogy :

ExempleEnergy balance for the conductive energetic field

Let us consider a conductive thread of conductivity , associated to a cylinder of axis and of radius .

A steady and uniform electric field reigns, both inside and outside the wire :

The wire is then crossed by volumic currents of uniform density :

The magnetic field created by this distribution is of this kind :

It can be computed using Ampere's circuital law.

By writing the total current crossing the transversal section of the wire, then :

  • If  :

  • If  :

Poynting's vector is :

Thus :

  • If  :

  • If  :

The general expression of electromagnetic energy conservation is reminded :

In the particular case of steady-state :

Physically, in steady-state, the power dissipated by Joule effect is evacuated outside the volume .

We compute the exiting flux of the Poynting vector through a cylinder of axis and radius .

  • If  :

  • If  :

We can recognize easily, in both cases, the power absorbed by Joule effect in the cylinder of radius considered (the volumic Joule dissipated power is ).

The previous conservation equation is verified.

Observation of magnetic energy of a coil (Vidéo d'Alain Le Rille, enseignant en CPGE au lycée Janson de Sailly, Paris)
Maxwell's equations in metals, skin effect
Skin effect in a metallic conductor plunged in a solenoid