electromagnetism - Relating Poyntings theorem to Lenz and Faraday's law?

In system's similar to a motor, where the armature begins to accelerate simultaneously there is induced $-\epsilon$ to reduce the applied current(hence the applied power $P(t)$ is also reduced), or others similar to that principle such as a rail gun, Lenz's law would state the conservation of energy in Faraday's law of induction, however, how does Poyntings theorm relate to such systems? To add to the conservation of energy?

Answer

First, let's start by defining some parameters:

• $\mu_{o}$ is the permeability of free space


• $\varepsilon_{o}$ is the permittivity of free space


• $\mathbf{E}$ is the 3-vector electric field


• $\mathbf{B}$ is the 3-vector magnetic field


• $\mathbf{S}$ is the 3-vector Poynting flux (also called the Poynting vector)


• $\mathbf{j}$ is the 3-vector electric current density



• $\partial_{\alpha}$ is the partial derivative with respect to parameter $\alpha$


• $q_{s}$ charge of particle species $s$


• $n_{s}$ number density of particle species $s$ (i.e., number per unit volume)


• $\mathbf{v}_{s}$ bulk flow velocity of particle species $s$

Poynting's theorem is defined mathematically (in differential form) as: $$ \partial_{t} \left( w_{B} + w_{E} \right) + \nabla \cdot \mathbf{S} = - \mathbf{j} \cdot \mathbf{E} \tag{1} $$ where $\partial_{t}$ is the partial time derivative, $w_{B} = B^{2}/\left( 2 \mu_{o} \right)$, $w_{E} = \varepsilon_{o} E^{2}/2$, $\mathbf{S} = \left( \mathbf{E} \times \mathbf{B} \right)/\mu_{o}$, and $\mathbf{j} = \sum_{s} \ q_{s} \ n_{s} \mathbf{v}_{s}$. $^{\mathbf{A}}$ We can often write Poynting's theorem in differential form because the volume over which one integrates (i.e., the surface through which $\mathbf{S}$ is leaving/entering) is generally arbitrary [e.g., see pages 258-264 in Jackson [1999]].

We can define Poynting's theorem in terms of physically significant phrases, like the following:

1. the time rate of change of the energy density of the electromagnetic fields; plus



2. the rate of electromagnetic energy flux flowing out of an arbitrary surface; equals


3. the energy lost due to momentum transfer between particles and fields.

We could just as easily describe 1. as the rate of energy transfer per unit volume, 2. as the power flowing out of a volume through a defined surface, and 3. as the rate of work done per unit volume on the charges in the volume element.

One thing to note is that when in differential form as in Equation 1, Poynting's theorem is one example of a continuity equation. All continuity equations are expressed as:

1. the time rate of change of a density; plus


2. the rate of flux flowing out of an arbitrary surface; equals


3. sources and losses.

In terms of units, a flux is just a density multiplied by a velocity. In simple (loose/careless) terms, the velocity gives the direction and rate while the density supplies the volume and number.

how does Poyntings theorm relate to such systems?

Poynting's theorem is, in short, a statement of the conservation of electromagnetic energy.

The $\left( \mathbf{j} \cdot \mathbf{E} \right)$ term in Equation 1 shows how energy is transformed from electromagnetic(particle mechanical) to particle mechanical(electromagnetic) energy.$^{\mathbf{B}}$ You can see this by recalling that one form for expressing power (i.e., energy per unit time) is given by: $$ P = \mathbf{F} \cdot \mathbf{v} $$ where $\mathbf{F}$ is some force acting on some object and $\mathbf{v}$ is the velocity of said object. When you look at the $\left( \mathbf{j} \cdot \mathbf{E} \right)$ term, you can see that it can be represented by: $$ \left( \mathbf{j} \cdot \mathbf{E} \right) = \mathbf{E} \cdot \left( \sum_{s} \ q_{s} \ n_{s} \mathbf{v}_{s} \right) \\ \sim \sum_{s} \ \frac{ \mathbf{F} }{ q_{s} } \cdot \left( q_{s} \ n_{s} \mathbf{v}_{s} \right) $$ where I have just rewritten $\mathbf{E}$ as $\mathbf{F}/q$ (from the Lorentz force). Then you can see that there is a term similar to $\mathbf{F} \cdot \mathbf{v}$ within $\left( \mathbf{j} \cdot \mathbf{E} \right)$. Thus, the $\left( \mathbf{j} \cdot \mathbf{E} \right)$ term is clearly a rate of change of energy per unit volume.

To add to the conservation of energy?

Poynting's theorem is part of the total conservation of energy of a given system. There are numerous ways to treat this and many can be too involved to go into here, but the simple answer is that it defines the conversion of particle energy to/from electromagnetic energy. For instance, one can use Poynting's theorem with the generalized Ohm's law [e.g., see page 572 in Jackson [1999]] when describing the transport properties (e.g., conductivity) of a given system.

So in a sense, yes, Poynting's theorem adds to the conservation of energy in that it is one part of that law.

A. I already converted the expression for current density to a macroscopic form. To see more details on the difference between micro- and macroscopic Maxwell's equations, see pages 248-258 in Jackson [1999].
B. Note that I have included heat (i.e., random kinetic energy) and bulk flow kinetic energy in my use of the term mechanical energy here.

• J.D. Jackson, Classical Electrodynamics, Third Edition, John Wiley & Sons, Inc., New York, NY, 1999.