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The Poynting Theorem and Poynting Vector
Do Not Necessarily Mean What They Are
Conventionally Said to Mean!

The Poynting Theorem is in the nature of a statement of the conservation of energy for a configuration consisting of electric and magnetic fields acting on charges. Consider a volume V with a surface ∂V. Then the conventional statement of the Poynting Theorem is as follows:

The time rate of change of electromagnetic energy within V plus the net energy flowing out of V through ∂V per unit time is equal to the negative of the total work done on the charges or charge distribution within V.

In symbols

The point-wise version:
−E·J = (dU/dt) + (c/4π)∇·(E×H)
where J is electrical current distribution.
The volume version:
−(dW/dt) = ∫V(dU/dt)dV + ∫∂V(c/4π)[(E×H)·n]dσ
where dσ is an element of the surface ∂V
and n is a unit normal to dσ.

The vector field quantity, (c/4π)(E×H) named the Poynting vector, is taken to be the energy flow per unit area and this is usually identified with electromagnetic radiation. The nature of the Poynting vector is somewhat different, as revealed by the following derivation.

The Dynamics of Electromagnetic Fields

The Maxwell equations describe the dynamics of an electric field and a magnet field. The electric field is specified by two vector fields E and D and the magnetic field by two vector fields B and H. The equations also involve the charge distribution and the current distribution. For a configuration of electric and magnetic fields in which there are no charges or currents the equations for the dynamics of the fields are, in terms of conventional notation,

(∂D/∂t) = c∇×H
(∂B/∂t) = −c∇×E

Since D=εE and B=μH, where ε is the dielectric of the material the fields are located in and μ is its permeability the above equations may be recast strictly in terms of E and H as

ε(∂E/∂t) = c∇×H
μ(∂H/∂t) = −c∇×E

The density of the energy stored in the fields is

U = (1/4π)[½εE·E + ½μH·H]
and hence
(dU/dt) = (1/4π)[εE·(∂E/∂t) + μH·(∂H/∂t)]

From the expressions for the dynamics of E and H the above formula reduces to

(dU/dt) = (c/4π)[E·(∇×H) −H·(∇×E)]

There is the vector identity

∇·(E×H) = H·(∇×E) − E·(∇×H)

A comparison of this identity with the RHS of the previous equation reveals that

(dU/dt) = −[(c/4π)∇·(E×H)]

The Poynting vector S is defined as (c/4π)(E×H). Thus

(dU/dt) = −∇·S

This means that the electric and magnetic fields are not in equilibrium unless of the divergence of the Poynting vector is equal to zero.

For any volume V

V(dU/dt)dV = −∫V(∇·S)dV

By means of Gauss' Divergence Theorem the RHS of the above equation can be put into the form


where ∂V is the boundary surface of the volume V, dσ is an element of that surface and n is a unit normal to that surface. Thus the Poynting vector S represents the dynamics of the electromagneticields. It can be construed to be a flow of energy, but that terminology is a bit misleading and represents the archaic notion of energy being like a fluid. If the fields move they take with them their energy. This would constitute an energy flow although primarily it is a movement of the fields themselves. In any case the flow of energy might or might not be the sinusoidal oscillation involved in electromagnet radiation. It may be simply a relaxation of an imbalance in the fields.


What the Poynting Theorem really says is that the time rate of change of the energy density of electromagnetic fields at a point is equal to the negative of the dispersion of the Poynting vector at that point. The energy flow at that point may be due solely to the movement of the fields carrying their energies with them. There may be no electromagnetic radiation at all involved.

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