Poynting vector encyclopedia topics

Interpretation

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

frac{partial u}{partial t} = - mathbf{nabla}cdotmathbf{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}^2right),

where

epsilon_0

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.

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.

Alternative form

The Poynting vector is often written as

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

where H the auxiliary magnetic field. This form appears in many textbooks and in Poynting's original paper . The two alternative definitions are equivalent in non-magnetic materials, where

B=mu_0 H

. However, in magnetic media, where

B=mu_0 mu_r H

, the two definitions are not equivalent. The definition in terms of

H

then no longer represents a flux of electromagnetic energy, but rather a combination of radiative energy and electrical currents in the material. Furthermore, in any real-world material, the constitutive relations are more complicated than a simple proportionality. Then, the energy density

u

which corresponds to this definition of

mathbf{S}

can no longer be given, and one cannot establish the energy continuity equation (Poynting's theorem) above. In consequence,

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

is usually not an appropriate energy flux vector.

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.

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 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.

The Poynting vector in plane 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.

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

Eleft(t,{mathbf r}right) = E_0,cosleft(omega,t- {mathbf k} cdot {mathbf r} right),

Bleft(t,{mathbf r}right) = B_0,cosleft(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^2left(omega t-{mathbf k} cdot {mathbf r}right) =

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

is ½, it follows that

leftlangle S rightrangle = 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

langlemathbf{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 Srangle}{c}.

Problems in certain cases

The common use of the Poynting vector as an energy flux rather than in the context of Poynting's theorem gives rise to controversial interpretions in cases where it is not used to describe electromagnetic radiation. Two examples are given below.

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

mathit{P} = mathit{V}mathit{I}

, where

mathit{V}

is the potential difference between the cable and ground,

mathit{I}

is the current carried by the cable. This power flows through the surrounding dielectric, and not through the cable itself.

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.

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}

. 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. 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.

References

Interpretation

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

frac{partial u}{partial t} = - mathbf{nabla}cdotmathbf{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}^2right),

where

epsilon_0

Alternative form

The Poynting vector is often written as

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

where H the auxiliary magnetic field. This form appears in many textbooks and in Poynting's original paper . The two alternative definitions are equivalent in non-magnetic materials, where

B=mu_0 H

. However, in magnetic media, where

B=mu_0 mu_r H

, the two definitions are not equivalent. The definition in terms of

H

u

which corresponds to this definition of

mathbf{S}

can no longer be given, and one cannot establish the energy continuity equation (Poynting's theorem) above. In consequence,

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

is usually not an appropriate energy flux vector.

Generalization

Examples and applications

The Poynting vector in a coaxial cable

The Poynting vector in plane waves

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

Eleft(t,{mathbf r}right) = E_0,cosleft(omega,t- {mathbf k} cdot {mathbf r} right),

Bleft(t,{mathbf r}right) = B_0,cosleft(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^2left(omega t-{mathbf k} cdot {mathbf r}right) =

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

is ½, it follows that

leftlangle S rightrangle = 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