Poynting's theorem

The Poynting's theorem (also Poynting Theorem ) describes the energy balance in the electrodynamics . This generalizes the law of conservation of energy to electromagnetic fields. Its formulation is attributed to the British physicist John Henry Poynting . In very simplified terms, it carries the statement that an electromagnetic field can do work if it becomes "weaker". Mathematically, like Maxwell's equations , it can be specified in both a differential and an integral notation. In the integral form it reads:

${\displaystyle \int _{V}{\vec {j}}\cdot {\vec {E}}\,\mathrm {d} V=-\int _{V}\left({\vec {\nabla }}\cdot {\vec {S}}+{\frac {\partial u}{\partial t}}\right)\,\mathrm {d} V}$

In which:

${\displaystyle u={\frac {1}{2}}({\vec {E}}\cdot {\vec {D}}+{\vec {B}}\cdot {\vec {H}})={\frac {1}{2}}(\varepsilon _{0}{\vec {E}}^{2}+\mu _{0}{\vec {H}}^{2})}$ electromagnetic energy density of the fields.

${\displaystyle {\vec {S}}={\vec {E}}\times {\vec {H}}}$ Poynting vector

${\displaystyle {\vec {j}}=\varrho \cdot {\vec {v}}}$ electric current density

${\displaystyle {\vec {E}},{\vec {H}}}$ electric and magnetic field strengths

He says that the change in energy in electromagnetic fields in a volume , ,, not only by the energy flow , occurs in or out of this volume (which would correspond to a continuity equation ), but can also be done through an exchange with other subsystems . The latter contribution is also called Joule heat , and states that energy is not preserved in electrodynamic subsystems, but can be exchanged with other subsystems, i.e. can be converted into kinetic, internal or chemical energy. This does not contradict the fact that the energy is retained in a closed system. The energy flow can be made more understandable by using Gaussian theorem in the integral form: ${\displaystyle V}$ ${\displaystyle \int _{V}{\frac {\partial u}{\partial t}}\,\mathrm {d} V}$ ${\displaystyle \int _{V}{\vec {\nabla }}\cdot {\vec {S}}\,\mathrm {d} V}$ ${\displaystyle \int _{V}{\vec {j}}\cdot {\vec {E}}\,\mathrm {d} V}$

${\displaystyle \int _{V}{\vec {j}}\cdot {\vec {E}}\,\mathrm {d} V=-\int _{\partial V}{\vec {S}}\cdot \mathrm {d} {\vec {A}}-\int _{V}{\frac {\partial u}{\partial t}}\,\mathrm {d} V}$

The surface integral then corresponds to the flow of the power density through the considered surface of the volume . ${\displaystyle V}$

Since only the divergence of is relevant, a rotation of an arbitrary function could in principle also be added to it, since it disappears under the effect of the divergence. The physical interpretation of as power flow is then no longer possible. So there are formally an infinite number of vector-valued functions that satisfy Poynting's theorem, but only can be obtained from the Maxwell equations and is therefore physically meaningful. ${\displaystyle {\vec {S}}}$ ${\displaystyle {\vec {S'}}={\vec {S}}+{\vec {\nabla }}\times {\vec {f}}}$ ${\displaystyle {\vec {S}}={\vec {E}}\times {\vec {H}}}$

Derivation

The starting point is the work that an electromagnetic field does on charge carriers per time and volume and the resulting power density:

${\displaystyle {\frac {\mathrm {d} ^{4}W}{\mathrm {d} V\,\mathrm {d} t}}={\frac {P}{V}}={\frac {{\vec {v}}\cdot q{\vec {E}}}{V}}=\varrho \cdot {\vec {v}}\cdot {\vec {E}}={\vec {j}}\cdot {\vec {E}}}$

It should be noted that the magnetic part of the field does no work, since the Lorentz force acts perpendicular to the direction of movement of the charge. But this applies Ampere's law : . What put up on ${\displaystyle {\vec {j}}={\vec {\nabla }}\times {\vec {H}}-{\frac {\partial {\vec {D}}}{\partial t}}}$

${\displaystyle {\vec {j}}\cdot {\vec {E}}={\vec {E}}\cdot ({\vec {\nabla }}\times {\vec {H}})-{\vec {E}}\cdot {\frac {\partial {\vec {D}}}{\partial t}}}$

leads. If you also draw the calculation rule for the divergence

${\displaystyle {\vec {\nabla }}\cdot ({\vec {E}}\times {\vec {H}})={\vec {H}}\cdot ({\vec {\nabla }}\times {\vec {E}})-{\vec {E}}\cdot ({\vec {\nabla }}\times {\vec {H}})}$

come up, so it turns out

${\displaystyle {\vec {j}}\cdot {\vec {E}}={\vec {H}}\cdot ({\vec {\nabla }}\times {\vec {E}})-{\vec {\nabla }}\cdot ({\vec {E}}\times {\vec {H}})-{\vec {E}}\cdot {\frac {\partial {\vec {D}}}{\partial t}}}$.

The rotation of the electric field can finally be expressed via the law of induction , which means ${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$

${\displaystyle {\vec {j}}\cdot {\vec {E}}=-{\vec {H}}\cdot {\frac {\partial {\vec {B}}}{\partial t}}-{\vec {\nabla }}\cdot ({\vec {E}}\times {\vec {H}})-{\vec {E}}\cdot {\frac {\partial {\vec {D}}}{\partial t}}}$

results. Here it only remains to summarize the equation with the help of the definition of the Poynting vector and the energy density, for which the following identities are required:

${\displaystyle {\vec {D}}=\varepsilon _{0}{\vec {E}}\qquad {\vec {B}}=\mu _{0}{\vec {H}}}$

and

${\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial }{\partial t}}\left({\frac {1}{2}}\varepsilon _{0}{\vec {E}}^{2}+{\frac {1}{2}}\mu _{0}{\vec {H}}^{2}\right)={\vec {E}}\cdot {\frac {\partial {\vec {D}}}{\partial t}}+{\vec {H}}\cdot {\frac {\partial {\vec {B}}}{\partial t}}}$

Which finally justifies the differential form of the sentence.

${\displaystyle {\vec {j}}\cdot {\vec {E}}=-{\vec {\nabla }}\cdot {\vec {S}}-{\frac {\partial u}{\partial t}}}$

Example: Ohmic resistance

If we consider a cylindrical conductor with radius and length , through which a current that is constant over time flows, whereby the voltage drops over the length of the conductor in proportion to the length. The conductor is therefore an ohmic resistance. The surface on which the Poynting vector, i.e. the electrical and magnetic field strength, is considered, is the surface of the cylinder. ${\displaystyle R}$ ${\displaystyle l}$ ${\displaystyle I}$ ${\displaystyle U}$ ${\displaystyle M}$

The amount of the electric field strength can be used approximately as with a plate capacitor. ${\displaystyle \left|{\vec {E}}\right|={\frac {U}{l}}}$

The magnetic field strength on the outer surface is that of a current-carrying conductor . ${\displaystyle \left|{\vec {H}}\right|={\frac {I}{2\pi R}}}$

The orientation of the electric field strength follows the length of the cylinder, the magnetic field strength the circumference. So they are always perpendicular to each other and lie in the area under consideration.

The magnitude of the Poynting vector is

${\displaystyle \left|{\vec {S}}\right|=\left|{\vec {E}}\times {\vec {H}}\right|={\frac {U\cdot I}{2\pi Rl}}}$.

The direction of the vector points into the conductor.

If you integrate the Poynting vector over the surface, you get the converted power.

${\displaystyle P=\oint _{M}{\vec {S}}\cdot \mathrm {d} {\vec {A}}=-{\frac {U\cdot I}{2\pi Rl}}\cdot 2\pi Rl=-U\cdot I}$

The negative sign takes into account the orientation of a closed surface, which is always outward.

The same considerations can be carried out using a battery, with a result that only differs in the sign of the power. This explains the energy flow for a simple circuit made up of a resistor and a battery. The battery emits the chemical energy stored in it in all spatial directions in the resulting electric and magnetic fields (only not in the current-carrying lines) and the resistance picks it up from all directions and then sets it, for example. B. in thermal energy. A battery is a source of electrical energy (which is stored in the fields), the resistance a sink.

literature

• John David Jackson: Classical Electrodynamics . 4th, revised edition. de Gruyter, Berlin 2006, ISBN 3-11-018970-4