Poynting vector

From Wikipedia, the free encyclopedia

In physics, the Poynting vector can be thought of as representing the energy flux (W/m2) of an electromagnetic field. It is named after its inventor John Henry Poynting. Oliver Heaviside independently co-discovered the Poynting vector. Usually, it is defined for free space as

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

where E is the electric field, H the auxiliary magnetic field, B the magnetic field, μ0 the magnetic constant. (All bold letters represent vectors.)

Contents

[edit] Interpretation in free space

The Poynting vector appears in the energy-conservation law[1], or Poynting's theorem,

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

where J is the current density and u is the electromagnetic energy density,

u = \frac{1}{2}\left(\epsilon_0 \mathbf{E}^2 + \frac{1}{\mu_0} \mathbf{B}^2\right),

where ε0 is the electric constant. The first term in the right-hand side represents the net electromagnetic energy flow into a small volume, while the second term represents the negative of work done by electrical currents that are not necessarily converted into electromagnetic energy.

[edit] The Poynting vector in electromagnetic waves

In a propagating sinusoidal electromagnetic plane wave of a fixed frequency, the Poynting vector oscillates, always pointing in the direction of propagation. 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.

[edit] 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.

[edit] 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 \langle\mathbf{S}\rangle divided by the speed of light in free space is the radiation pressure exerted by an electromagnetic wave on the surface of a target:

P_{rad}=\frac{\langle S\rangle}{c}.

[edit] Examples

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 the core conductor was replaced by a wire having significant resistance, then the Poynting vector would become tilted toward that wire, indicating that energy flows from the electromagnetic field into the wire, producing resistive Joule heating in the wire.

[edit] Problems in certain cases

The definition of the Poynting vector gives rise to controversial interpretations in certain cases:

[edit] DC Power flow in a concentric cable

Application of Poynting's Theorem to a concentric cable carrying DC current leads to the correct power transfer equation P = VI, where V is the potential difference between the cable and ground, I is the current carried by the cable. This power flows through the surrounding dielectric, and not through the cable itself. [2]

However, it is also known that power cannot be radiated without accelerated charges, i.e. time varying currents. Since we are considering DC (time invariant) currents here, radiation is not possible. This has led to speculation that Poynting Vector may not represent the power flow in certain systems.[3][4]

[edit] Independent E and B fields

Independent static \mathbf{E} and \mathbf{B} fields do not result in power flows along the direction of \mathbf{E} \times \mathbf{B}, as predicted by the Poynting's theorem.

For example, application of Poynting's Theorem to a bar magnet, on which an electric charge is present, leads to seemingly absurd conclusion that there is a continuous circulation of energy around the magnet.[2] However, there is no divergence of energy flow, or in layman's terms, energy that enters given unit of space equals the energy that leaves that unit of space, so there is no net energy flow into the given unit of space.

[edit] Generalization

Poynting vector represents particular case of energy flux vector for electromagnetic field. However, any type of energy has its direction of movement in space, as well as its density. The first one is represented by Umov-Poynting vector discovered by Nikolay Umov in 1874 for liquid and elastic media in completely generalized view.

[edit] References

  1. ^ John David Jackson (1998). Classical electrodynamics, Third Edition, New York: Wiley. ISBN 047130932X. 
  2. ^ a b Jordan, Edward & Balmain, Keith (2003), Electromagnetic Waves and Radiating Systems (Second ed.), New Jersey: Prentice-Hall, ISBN 81-203-0054-8, <http://worldcat.org/isbn/8120300548> 
  3. ^ Jeffries, Clark (Sep., 1992). "A New Conservation Law for Classical Electrodynamics" (PDF). . Society for Industrial and Applied Mathematics (SIAM Review) Retrieved on 2008-03-04.
  4. ^ Robinson, F. N. H. (Dec., 1994). "Poynting's Vector: Comments on a Recent Paper by Clark Jeffries". . Society for Industrial and Applied Mathematics (SIAM Review)

[edit] Further reading

[edit] See also