Poynting's theorem

In electrodynamics, Poynting's theorem is a statement of conservation of energy for the electromagnetic field, in the form of a partial differential equation, due to the British physicist John Henry Poynting.^[1] Poynting's theorem is analogous to the work-energy theorem in classical mechanics, and mathematically similar to the continuity equation, because it relates the energy stored in the electromagnetic field to the work done on a charge distribution (i.e. an electrically charged object), through energy flux.

Statement

General

In words, the theorem is an energy balance:

The rate of energy transfer (per unit volume) from a region of space equals the rate of work done on a charge distribution plus the energy flux leaving that region.

A second statement can also explain the theorem - "The decrease in the electromagnetic energy per unit time in a certain volume is equal to the sum of work done by the field forces and the net outward flux per unit time".

Mathematically, this is summarised in differential form as:

$-\frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S}+\mathbf{J}_f\cdot\mathbf{E}$

where ∇•S is the divergence of the Poynting vector (energy flow) and J_f•E is the rate at which the fields do work on a charged object (J_f is the free current density corresponding to the motion of charge, E is the electric field, and • is the dot product). The energy density u is given by:^[2]

$u = \frac{1}{2}\left(\mathbf{E}\cdot\mathbf{D} + \mathbf{B}\cdot\mathbf{H}\right)$

in which D is the electric displacement field, B is the magnetic flux density and H the magnetic field strength. Since only some of the charges are free to move, and the D and H fields exclude the "bound" charges and currents in the charge distribution (by their definition), one obtains the free current density J_f in the Poynting theorem, rather than the total current density J.

Using the divergence theorem, Poynting's theorem can be rewritten in integral form:

$-\frac{\partial}{\partial t} \int_V u dV=$$\scriptstyle \partial V$$\mathbf{S}\cdot d\mathbf{A} + \int_V\mathbf{J}\cdot\mathbf{E}dV$

where $\partial V \!$ is the boundary of a volume V. The shape of the volume is arbitrary but fixed for the calculation.

Electrical engineering

In electrical engineering context the theorem is usually written with the energy density term u expanded in the following way, which resembles the continuity equation:

$\nabla\cdot\mathbf{S} + \epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t} + \frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t} + \mathbf{J}\cdot\mathbf{E} = 0,$

where

• ε₀ is the electric constant and μ₀ is the magnetic constant.
• $\epsilon_0 \mathbf{E}\cdot\frac{\partial \mathbf{E}}{\partial t}$ is the density of reactive power driving the build-up of electric field,
• $\frac{\mathbf{B}}{\mu_0}\cdot\frac{\partial\mathbf{B}}{\partial t}$ is the density of reactive power driving the build-up of magnetic field, and
• $\mathbf{J}\cdot\mathbf{E}$ is the density of electric power dissipated by the Lorentz force acting on charge carriers.

Derivation

While conservation of energy and the Lorentz force law can derive the general form of the theorem, Maxwell's equations are additionally required to derive the expression for the Poynting vector and hence complete the statement.

Poynting's theorem

Considering the statement in words above - there are three elements to the theorem, which involve writing energy transfer (per unit time) as volume integrals:^[3]

1. Since u is the energy density, integrating over the volume of the region gives the total energy U stored in the region, then taking the (partial) time derivative gives the rate of change of energy:

   $U=\int_V u dV \ \rightarrow \ \frac{\partial U}{\partial t} = \frac{\partial}{\partial t} \int_V u dV = \int_V \frac{\partial u}{\partial t} dV .$

2. The energy flux leaving the region is the surface integral of the Poynting vector, and using the divergence theorem this can be written as a volume integral:

   $\scriptstyle \partial V$$\mathbf{S}\cdot d\mathbf{A}=\int_V\nabla\cdot \mathbf{S} dV .$

3. The Lorentz force density f on a charge distribution, integrated over the volume to get the total force F, is

   $\mathbf{f} = \rho\mathbf{E}+\mathbf{J}\times\mathbf{B} \ \rightarrow \ \int_V \mathbf{f} dV = \mathbf{F} = \int_V (\rho\mathbf{E}+\mathbf{J}\times\mathbf{B} )dV ,$

   where ρ is the charge density of the distribution and v its velocity. Since $\mathbf{J} = \rho \mathbf{v}$, the rate of work done by the force is

   $\mathbf{F}\cdot\frac{d \mathbf{r}}{dt} = \mathbf{F}\cdot \mathbf{v} = \int_V (\rho\mathbf{E}\cdot\mathbf{v}+\rho\mathbf{v}\times\mathbf{B}\cdot \mathbf{v} )dV \ \rightarrow \ \mathbf{F}\cdot \mathbf{v} = \int_V \mathbf{E}\cdot \mathbf{J}dV .$

So by conservation of energy, the balance equation for the energy flow per unit time is the integral form of the theorem:

$-\int_V\frac{\partial u}{\partial t}dV = \int_V\nabla\cdot\mathbf{S}dV+\int_V\mathbf{J}\cdot\mathbf{E}dV,$

and since the volume V is arbitrary, this is true for all volumes, implying

$- \frac{\partial u}{\partial t} = \nabla\cdot\mathbf{S} + \mathbf{J}\cdot\mathbf{E},$

which is Poynting's theorem in differential form.

Poynting vector

From the theorem, the actual form of the Poynting vector S can be found. The time derivative of the energy density (using the product rule for vector dot products) is

$\frac{\partial u}{\partial t} = \frac{1}{2} \left(\mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t} + \mathbf{D}\cdot\frac{\partial \mathbf{E}}{\partial t} + \mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t} + \mathbf{B}\cdot\frac{\partial \mathbf{H}}{\partial t}\right)= \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t} + \mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t},$

using the constitutive relations

$\mathbf{D} = \epsilon_0 \mathbf{E},\quad \mathbf{B} = \mu_0 \mathbf{H}.$

The partial time derivatives suggest using two of Maxwell's Equations. Taking the dot product of the Maxwell–Faraday equation with H:

$\frac{\partial \mathbf{B}}{\partial t} = - \nabla \times \mathbf{E} \ \rightarrow \ \mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t} = - \mathbf{H}\cdot\nabla \times \mathbf{E},$

next taking the dot product of the Maxwell–Ampère equation with E:

$\frac{\partial \mathbf{D}}{\partial t} + \mathbf{J} = \nabla \times \mathbf{H} \ \rightarrow \ \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t} + \mathbf{E}\cdot\mathbf{J} = \mathbf{E}\cdot\nabla \times \mathbf{H}.$

Collecting the results so far gives:

$\begin{align} -\nabla\cdot\mathbf{S} & = \frac{\partial u}{\partial t} +\mathbf{J}\cdot\mathbf{E} \\ & = \left(\mathbf{H}\cdot\frac{\partial \mathbf{B}}{\partial t} + \mathbf{E}\cdot\frac{\partial \mathbf{D}}{\partial t}\right) + \mathbf{J}\cdot\mathbf{E} \\ & = \mathbf{E}\cdot\nabla \times \mathbf{H} - \mathbf{H}\cdot\nabla \times \mathbf{E}, \\ \end{align}$

then, using the vector calculus identity:

$\nabla \cdot\mathbf{E} \times \mathbf{H}=\mathbf{H}\cdot\nabla \times \mathbf{E} - \mathbf{E}\cdot\nabla \times \mathbf{H},$

gives an expression for the Poynting vector:

$\mathbf{S} =\mathbf{E} \times \mathbf{H} ,$

which physically means the energy transfer due to time-varying electric and magnetic fields is perpendicular to the fields.

Alternative forms

It is possible to derive alternative versions of Poynting's theorem.^[4] Instead of the flux vector E × B as above, it is possible to follow the same style of derivation, but instead choose the Abraham form E × H, the Minkowski form D × B, or perhaps D × H. Each choice represents the response of the propagation medium in its own way: the E × B form above has the property that the response happens only due to electric currents, while the D × H form uses only (fictitious) magnetic monopole currents. The other two forms (Abraham and Minkowski) use complementary combinations of electric and magnetic currents to represent the polarization and magnetization responses of the medium.

Generalization

The mechanical energy counterpart of the above theorem for the electromagnetic energy continuity equation is

$\frac{\partial}{\partial t} u_m(\mathbf{r},t) + \nabla\cdot \mathbf{S}_m (\mathbf{r},t) = \mathbf{J}(\mathbf{r},t)\cdot\mathbf{E}(\mathbf{r},t),$

where u_m is the (mechanical) kinetic energy density in the system. It can be described as the sum of kinetic energies of particles α (e.g., electrons in a wire), whose trajectory is given by r_α(t):

$u_m(\mathbf{r},t) = \sum_{\alpha} \frac{m_{\alpha}}{2} \dot{r}^2_{\alpha} \delta(\mathbf{r}-\mathbf{r}_{\alpha}(t)),$

where S_m is the flux of their energies, or a "mechanical Poynting vector":

$\mathbf{S}_m (\mathbf{r},t) = \sum_{\alpha} \frac{m_{\alpha}}{2} \dot{r}^2_{\alpha}\dot{\mathbf{r}}_{\alpha} \delta(\mathbf{r}-\mathbf{r}_{\alpha}(t)).$

Both can be combined via the Lorentz force, which the electromagnetic fields exert on the moving charged particles (see above), to the following energy continuity equation or energy conservation law:^[5]

$\frac{\partial}{\partial t}\left(u_e + u_m\right) + \nabla\cdot \left( \mathbf{S}_e + \mathbf{S}_m\right) = 0,$

covering both types of energy and the conversion of one into the other.

Notes

1. Poynting, J. H. (1884). "On the Transfer of Energy in the Electromagnetic Field". Philosophical Transactions of the Royal Society of London 175: 343–361. doi:10.1098/rstl.1884.0016.
2. Electromagnetism (2nd Edition), I.S. Grant, W.R. Phillips, Manchester Physics, John Wiley & Sons, 2008, chapters 2 and 6, ISBN 9-780471-927129
3. Introduction to Electrodynamics (3rd Edition), D.J. Griffiths, Pearson Education, Dorling Kindersley, 2007, p.364, ISBN 81-7758-293-3
4. Kinsler, P.; Favaro, A.; McCall M.W. (2009). "Four Poynting theorems". European Journal of Physics 30 (5): 983. arXiv:0908.1721. Bibcode:2009EJPh...30..983K. doi:10.1088/0143-0807/30/5/007.
5. Richter, E.; Florian, M.; Henneberger, K. (2008). "Poynting's theorem and energy conservation in the propagation of light in bounded media". Europhysics Letters 81 (6): 67005. arXiv:0710.0515. Bibcode:2008EL.....8167005R. doi:10.1209/0295-5075/81/67005.

External links

• Eric W. Weisstein "Poynting Theorem" From ScienceWorld – A Wolfram Web Resource.