Poynting vector

In physics, the Poynting vector can be thought of as representing the energy flux (in W/m²) of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside and Nikolay Umov independently co-invented the Poynting vector. In Poynting's original paper and in many textbooks it is defined as :

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

which is often called the Abraham form; here E is the electric field and H the auxiliary magnetic field. :

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

where J_f is the current density of free charges and u is the electromagnetic energy density, :

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

where B is the magnetic field and D the electric displacement field.

The first term in the right-hand side represents the net electromagnetic energy flow into a small volume, while the second term represents the subtracted portion of the work done by free electrical currents that are not necessarily converted into electromagnetic energy (dissipation, heat). In this definition, bound electrical currents are not included in this term, and instead contribute to S and u.

Note that u can only be given if linear, nondispersive and uniform materials are involved, i.e., if the constitutive relations can be written as : $\mathbf{D} = \epsilon\mathbf{E}, \;\;\; \mathbf{H} = \mathbf{B}/\mu$ where ε and μ are constants (which depend on the material through which the energy flows), called the permittivity and permeability, respectively, of the material.

This practically limits Poynting's theorem in this form to fields in vacuum. A generalization to dispersive materials is possible under certain circumstances at the cost of additional terms and the loss of their clear physical interpretation.

The Poynting vector is usually interpreted as an energy flux, but this is only strictly correct for electromagnetic radiation. The more general case is described by Poynting's theorem above, where it occurs as a divergence, which means that it can only describe the change of energy density in space, rather than the flow.

Formulation in terms of microscopic fields

In some cases, it may be more appropriate to define the Poynting vector S as :

\mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B},

where

\mu_0

is the magnetic constant. It can be derived directly from Maxwell's equations in terms of total charge and current and the Lorentz force law only.

The corresponding form of Poynting's theorem is : $\frac{\partial u}{\partial t} + \nabla\cdot\mathbf{S} = - \mathbf{J}\cdot\mathbf{E},$ where $\mathbf{J}$ is the total current density and the energy density $u$ is : $u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}^2 + \frac{\mathbf{B}^2}{\mu_0}\right)$ (with the electric constant $\epsilon_0$ ).

The two alternative definitions of the Poynting vector are equivalent in vacuum or in non-magnetic materials, where $\mathbf{B}=\mu_0 \mathbf{H}$ . In all other cases, they differ in that $\mathbf{S}=1/\mu_0 \mathbf{E}\times\mathbf{B}$ and the corresponding u are purely radiative, since the dissipation term, $-\mathbf{J}\cdot\mathbf{E}$ , covers the total current, while the definition in terms of $\mathbf{H}$ has contributions from bound currents which then lack in the dissipation term.

Invariance to adding a curl of a field

Since the Poynting vector only occurs in Poynting's theorem as a divergence

\nabla\cdot\mathbf{S}

, the Poynting vector is arbitrary to the extent that one can add a field curl

\nabla\times\mathbf F

, since

\nabla\cdot(\nabla\times\mathbf F)=0

for an arbitrary field F. Doing so is not common, though, and will lead to inconsistencies in a relativistic description of electromagnetic fields in terms of the stress-energy tensor.

Generalization

The Poynting vector represents the particular case of an energy flux vector for electromagnetic energy. However, any type of energy has its direction of movement in space, as well as its density, so energy flux vectors can be defined for other types of energy as well, e.g., for mechanical energy. The Umov-Poynting vector discovered by Nikolay Umov in 1874 describes energy flux in liquid and elastic media in a completely generalized view.

Time-averaged Poynting vector

For time-harmonic electromagnetic fields, the average power flow over time can be found as follows, :

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

= \operatorname{Re}\left(\mathbf{E} e^{j\omega t}\right) \times \operatorname{Re}\left(\mathbf{H} e^{j\omega t}\right)

= \frac{1}{2}\left(\mathbf{E} e^{j\omega t} + \mathbf{E}^* e^{-j\omega t}\right) \times \frac{1}{2}\left(\mathbf{H} e^{j\omega t} + \mathbf{H}^* e^{-j\omega t}\right)

= \frac{1}{4}\left(\mathbf{E} \times \mathbf{H}^* + \mathbf{E}^* \times \mathbf{H} + \mathbf{E} \times \mathbf{H} e^{2j\omega t} + \mathbf{E}^* \times \mathbf{H}^* e^{-2j\omega t}\right)

= \frac{1}{4}\left(\mathbf{E} \times \mathbf{H}^* + \left(\mathbf{E} \times \mathbf{H}^*\right)^* + \mathbf{E} \times \mathbf{H} e^{2j\omega t} + \left(\mathbf{E} \times \mathbf{H} e^{2j\omega t}\right)^*\right)

= \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}^*\right) + \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H} e^{2j\omega t}\right)

The average over time is given as :

\bar{\mathbf{S}} = \frac{1}{T}\int_0^T \mathbf{S}(t)dt = \frac{1}{T}\int_0^T \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}^*\right) + \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H} e^{2j\omega t}\right) dt

The second term is a sinusoidal curve (

\operatorname{Re}\left(e^{2j\omega t}\right) = \operatorname{cos}2\omega t

) whose average will be zero, which gives :

\bar{\mathbf{S}} = \frac{1}{2}\operatorname{Re}\left(\mathbf{E} \times \mathbf{H}^*\right)

Examples and applications

The Poynting vector in a coaxial cable

For example, the Poynting vector within the dielectric insulator of a coaxial cable is nearly parallel to the wire axis (assuming no fields outside the cable) – so electric energy is flowing through the dielectric between the conductors. If a conductor has significant resistance, then, near the surface of that conductor, the Poynting vector would be tilted toward and impinge upon the conductor. Once the Poynting vector enters the conductor, it is bent to a direction that is almost perpendicular to the surface. This is a consequence of Snell's law and the very slow speed of light inside a conductor. Inside the conductor, the Poynting vector represents energy flow from the electromagnetic field into the wire, producing resistive Joule heating in the wire.

The Poynting vector in plane waves

In a propagating sinusoidal electromagnetic plane wave of a fixed frequency, the Poynting vector always points in the direction of propagation while oscillating in magnitude. The time-averaged magnitude of the Poynting vector is :

\langle S \rangle = \frac{1}{2 \mu_0 c} E_0^2 = \frac{\epsilon_0 c}{2}  E_0^2,

where

\ E_0

is the maximum amplitude of the electric field and

\ c

is the speed of light in free space. This time-averaged value is also called the irradiance or intensity I.

Derivation

In an electromagnetic plane wave,

\mathbf{E}

and

\mathbf{B}

are always perpendicular to each other and the direction of propagation. Moreover, their amplitudes are related according to :

B_0 = \frac{E_0}{c},

and their time and position dependences are :

E\left(t,{\mathbf r}\right) = E_0\,\cos\left(\omega\,t- {\mathbf k} \cdot {\mathbf r} \right),

B\left(t,{\mathbf r}\right) = B_0\,\cos\left(\omega\,t- {\mathbf k} \cdot {\mathbf r} \right),

where

\ \omega

is the frequency of the wave and

\mathbf{k}

is wave vector. The time-dependent and position magnitude of the Poynting vector is then :

S(t) = \frac{1}{\mu_0} E_0\,B_0\,\cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r}\right) =
   \frac{1}{\mu_0 c} E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right) =
   \epsilon_0 c E_0^2 \cos^2\left(\omega t-{\mathbf k} \cdot {\mathbf r} \right).

In the last step, we used the equality

\epsilon_0\,\mu_0 = {c}^{-2}

. Since the time- or space-average of

\cos^2\left(\omega\,t-{\mathbf k} \cdot {\mathbf r}\right)

is ½, it follows that :

\left\langle S \right\rangle = \frac{\epsilon_0 c}{2} E_0^2.

Poynting vector and radiation pressure

S divided by the square of the speed of light in free space is the density of the linear momentum of the electromagnetic field. The time-averaged intensity