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Poynting vector

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 as

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

where E is the electric field, H the auxiliary magnetic field, B the magnetic field, $μ 0$ the permeability of vacuum, and $μ r$ the dimensionless relative permeability of the surrounding medium. (All bold letters represent vectors.)

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 e/m field into the wire, producing resistive Joule heating in the wire.

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1 Interpretation
2 The Poynting vector in electromagnetic waves
- 2.1 Derivation
3 Radiation pressure
4 References
5 See also

Interpretation

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 $ε 0$ is the permittivity of free space and $ε r$ the relative dielectric constant of the medium. The first term in the right-hand side represents the net electromagnetic energy flow into a small volume, while the second term represents negative of the work done by electrical currents that are not necessarily converted into electromagnetic energy.

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

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

where $E 0$ is the maximum amplitude of the electric field and c is the speed of light. 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 $ω$ 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 $\frac{1}{2}$ , it follows that

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

Radiation pressure

S divided by the square of the speed of light in a vacuum 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 a vacuum is the radiation pressure exerted by an electromagnetic wave on the surface of a target:

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

References

"Poynting Vector" from ScienceWorld (A Wolfram Web Resource) by Eric W. Weisstein
Richard Becker, Electromagnetic Fields and Interactions, Dover Publications, 1964

^ J.D. Jackson (1975). Classical Electrodynamics, 2nd Ed..

Poynting vector

Additional recommended knowledge

Daily Sensitivity Test

What is the Correct Way to Check Repeatability in Balances?

Better weighing performance in 6 easy steps

Contents

Interpretation

The Poynting vector in electromagnetic waves

Derivation

Radiation pressure

References

See also

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