Poynting vector

The Poynting vector ( named after the British physicist John Henry Poynting ) marks in electrodynamics ( a branch of physics ) the density and the direction of energy transport (energy flux density ) of an electromagnetic wave. The term of the energy flow is identical to the physical concept of power, the term energy flux density is therefore equivalent to the power density.

Mathematical Description

Of Poynting vector is a three-component vector, which points in the spatial direction of the energy flow. Its amount is equivalent to a part of the power density (or intensity ) of the shaft (the energy which is perpendicular through a unit area passing the Poynting vector per unit time) and on the other hand, the pulse density of the shaft ( of the pulse, that is stored per unit volume in the electromagnetic field ) is multiplied by the square of the speed of light. The magnitude of the Poynting vector thus has the dimension

Or equivalent

The Poynting vector is commonly referred to and is in electrodynamics in transverse electromagnetic wave ( TEM wave ) the cross product of the electric field strength and magnetic field strength.

Applies in a vacuum

The power density of a TEM wave is given by

Wherein the characteristic impedance of vacuum.

In the above equations, the field quantities depend on time meant. For the time average of the power density over a period of time applies with

Here, the effective value and the amplitude of a sinusoidal electric field strength.

In isotropic optical media, the Poynting vector is parallel to the wave vector. In anisotropic optical media, for example in birefringent crystals, this is not true in general.

The Poynting vector describes three of the ten independent components of the energy-momentum tensor of the electromagnetic field in the theory of relativity.

Application

The Poynting vector is in the set of Poynting, a conservation law of electrodynamics is considered.

Examples

Energy propagation in the coaxial cable

The typical operation of a coaxial conductor takes place at wavelengths that are greater than the diameter of the coaxial conductor. In this frequency range, which typically extends from 0 Hz to the single digits in the GHz range, the energy in the coaxial line spreads out as so-called TEM fundamental mode. The corresponding field pattern will look like in the picture.

In an ideal-typical analysis of the Poynting vector takes exclusively in the area between the outer conductor and the inner conductor to a non-zero value. In metallic conductors, even the Poynting vector vanishes, since the electric field strength is zero. Outside of the coaxial conductor of the Poynting vector vanishes, since the magnetic field vector is zero. The reason for the disappearance of the magnetic field outside the conductor is that the effect of the electric currents cancel in the inner and outer conductors against each other. According to the set of the Poynting Poynting vector denotes the direction of propagation of the electric power. Because of the disappearance of the electric field strength in the metal of the Poynting vector is exactly in the longitudinal direction of the coaxial conductor. This means that the energy spread in the coaxial conductor takes place only in the dielectric, and is ideal for typical viewing in the longitudinal direction of the conductor. Since the set of Poynting can be derived starting from the general field equations without any restriction on the frequency domain ( see Simonyi ), this statement also applies to the transmission of electrical power with DC voltages and currents.

The behavior of a resistive conductor can be explained in the box model. The following presentation is based on the coaxial shown in the image: Has the metallic conductor a nonzero finite resistance, the result by the current flow in the conductor according to Ohm's law an electric field. This field displays in the inner conductor in the longitudinal direction (x ) of the conductor, and is in the jacket head in the opposite direction ( o) directed. The altered field distribution causes the electric field in the dielectric is replaced by a component in the longitudinal direction. The orthogonal to E and H Poynting vector S has consequently a radial field component, which describes the transition of the energy loss in the metal.

Poynting vector in static fields

The consideration of the Poynting vector in static fields shows the relativistic nature of the Maxwell equations and allows a better understanding of the magnetic component of the Lorentz force. To illustrate the accompanying picture is considered that describes the Poynting vector in a cylindrical capacitor, which is located in a signal generated by a permanent magnet H field. Although only static electric and magnetic fields are present, the calculation of the Poynting vector results in a current flowing in the circuit electromagnetic energy, which can be assigned to an angular momentum. The rotational pulse contained in the circular energy flow, the cause of the discharge occurring in the magnetic component of the Lorentz force. During the unloading, the angular momentum present in the flow of energy is reduced and delivered to the charges of the discharge current. The seemingly nonsensical and paradoxical result of the circular flow of energy thus proving almost as necessary to be the law of momentum conservation requirements. ( Other static Examples: Feynman )