﻿ The Derivation and Explanation of the Poynting Theorem
San José State University

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Thayer Watkins
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Derivation and Explanation
of the Poynting Theorem

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 S. Then

#### the time rate of change of electromagnetic energy within V plus the net energy flowing out of V through S per unit time is equal to the negative of the total work done on the charges within V.

Consider first a single particle of charge q traveling with a velocity vector v. Let E and B be electric and magnetic fields external to the particle; i.e., E and B do not include the electric and magnetic fields generated by the moving charged particle. The force on the particle is given by the Lorentz formula

#### F = q(E + v×B)

The work done by the electric field on that particle is equal to qv·E. The work done by the magnetic field on the particle is zero because the force due to the magnetic field is perpendicular to the velocity vector v.

For a vector field of current density J the work done on the charges within a volume V is

#### ∫VJ·EdV

For a single particle of charge q traveling with velocity v the above quantity reduces to qv·E.

One form of the Ampere-Maxwell's Law says that

#### J = (c/4π)∇×H − (1/4π)(∂D/∂t)

When the RHS of the above is substituted for J the work done by the external fields on the charges within a volume V is

#### (1/4π)∫V[cE·(∇×H) − E·(∂D/∂t)]dV

There is a vector identity

This means that

#### E·(∇×H) = − ∇·(E×H) + H·(∇×E)

When this expression is substituted into the expression for the rate at which work is being done the result is

#### ∇×E = −(1/c)(∂B/∂t)

When Faraday's law is taken into account the previous equation can be expressed as:

#### ∫VJ·EdV = (−1/4π)∫V[c∇·(E×H) + E·(∂D/∂t) + H·(∂B/∂t)]dV

The total energy density U of the fields at a point is

#### U = (1/8π)(E·D + B·H)

where D=εE and H=(1/μ)B and ε and μ, called the dielectric and permabiity, respectively, are properties of the material in which the fields are located. The dielectric and permability are independent of the location.

This means that

#### U = (1/8π)(εE·E + (1/μ)B·B) and thus (∂U/∂t) = (1/4π)(εE·(∂E/∂t) + (1/μ)B·(∂B/∂t)) which is equivalent to (∂U/∂t) = (1/4π)(E·(∂D/∂t) + B·(∂H/∂t))

The RHS of this latter expression occurs in a previous expression so that

#### −∫VJ·EdV = ∫V[(∂U/∂t) + (c/4π)∇·(E×H)]dV

It is convenient to define a vector P, known as the Poynting vector for the electrical and magnetic fields, such that

#### P = (c/4π)(E×H)

The previous equation then becomes

#### −∫VJ·EdV = ∫V[(∂U/∂t) + ∇·P]dV

By Gauss' Divergence Theorem

#### ∫V(∇·P)dV = ∫Sn·PdS

where S is the surface of the volume V and n is the unit normal to the surface element dS. The vector P has the dimensions of energy×time per unit area. Thus ∫Sn·PdS is the net flow of energy out of the volume V.

The above means that work done by the electric and magnetic fields on the charges within a volume must match the rate of decrease of the energy of the fields within that volume and the net flow of energy into the volume. The big question is what does the net flow of energy into the volume correspond to physically. One possibility is that it might correspond to electromagnetic radiation. The above equation can also be stated as the negative of the work done on the charges within a volume must be equal to the increase in the energy of the electric and magnetic fields within the volume plus the net flow of energy out of the volume.

There is a major problem with the Poynting vector P; it is independent of the charges involved. It is the same whether there is one charge or one hundred million charges, or for that matter, zero charges. It can change with time but only as a result of the changes in the electric and magnetic fields.

Usually any difference between the change in energy and the work done is the energy of radiation. This is what is universally presumed in the case of the Poynting theorem, but the empirical evidence is that this cannot be so. If the Poynting vector corresponded to radiation then if a permanent magnet was placed in the vicinity of a body charged with static electricity the combination should glow and is that is not the case.

The Poynting vector is completely independent of the charges and their velocities in the volume being considered. In a word it is exogenous. The charges and their velocities are also exogenous. It is the rate of change of the energy stored in the fields that is endogenous. The Poynting theorem should read rate of change of energy in the fields = negative of work done by the fields on the charged particles minus the Poynting vector term.

However in the case of a permanent magnet and static electric charge the fields cannot change. Charged particles impinging upon an electric and magnetic field would experience work of them. The compensating change in momentum and energy would occur in the bodies holding the electric and magnetic fields. The charged particles hitting the electric and magnetic fields would induce a reaction as though they hit the magnet and charged body which creates the fields.

The dimensions of the Poynting vector term are energy per unit area per unit time. This is what would be expected if there were radiation generated in the volume. But the fact that the Poynting vector is exogenous means that without any charged particles at all being involved there would be radiation generated. The amount of radiation generated is fixed and no matter how many charged particles are injected into the volume at whatever velocities the same amount of radiation would be generated.

So the Poynting vector term apparently does not correspond to radiation. It is a puzzle as to what it does correspond to but there is no possibility that it corresponds to radiation.

## The Differential Form of the Poynting Theorem

Since the volume element is arbitrary the above equation implies that

#### (∂U/∂t) + ∇·P = −E·J

The interpretation of the term ∇·P is also problematical. It has a sign but it does not have a direction. It also is independent of the charge distribution, in this case J. In another study the case will be made that ∇·P is the time rate of change of the energy resulting from the interaction of the electrical and magnet field.