Poynting's theorem

The set of Poynting (also called Poyntingtheorem ) represents a conservation law in electrodynamics dar. Thus, the energy conservation law is generalized to electromagnetic fields. Its formulation is the British physicist John Henry Poynting attributed. Simplistically, it carries within it the statement that an electromagnetic field can do work when there " weaker " is. Mathematically it can, as well as Maxwell's equations, both in a differential as well as an integral notation are given. In the integral form, it is:

Where:

He says that the performance of a field is equal to the outlet of the energy flow and the decrease in the energy field. Thus it is comparable to the energy conservation law. This can be made clear when applying the Gauss theorem in the integral form:

The surface integral then corresponds to the flow of power density due to the viewed surface of the volume.

Since only the divergence is relevant, in principle, also a rotation of an arbitrary function may be added to it, since they disappear under the effect of divergence. The physical interpretation of the power flow is then no longer possible. So there is formally an infinite number of vector-valued functions that satisfy the set of Poynting, but only can be obtained from the Maxwell equations and is therefore physically meaningful.

Derivation

The starting point is the work performed by an electromagnetic field of charge carriers per unit time and volume:

It should be noted that the magnetic portion of the field verrichet not work, because the Lorentz force is perpendicular to the direction of movement of the charge. Now apply the Ampere's law: . What used on top of

Leads. One draws nigh nor the calculation rule for the divergence

Approach, we obtain

The rotation of the electric field can be expressed on the induction law, which we in

Were arriving. It remains to summarize only with the help of the definition of the Poynting vector and the energy density equation, which must be added the following identities are needed:

And

What, finally, the differential form of the sentence would be justified.

Example: ohmic resistance

Note: In this example, the CGS system of units used.

We consider a long cylindrical conductor of radius and length. The head is drained by the time-constant current, the length of the conductor drops the voltage U. The head is thus an ohmic resistance. Here, it is assumed that the spec. Resistance is the same everywhere.

We obtain the electric field.

The magnetic flux density outside the conductor we calculate with the Ampère law in integral form. The edge of the surface is ( with, the whole cylinder is enclosed ), a circle with radius, so that follows the Stokes' theorem for right away:

The field lines rotate the head by the corkscrew rule.

The Poynting vector is. He points into the conductor.

We define a closed surface around the entire head piece. The surface integral over the surface, which defines the whole energy loss per unit time.

The statement of the theorem of Poynting is that a source such as must compensate for this loss, a battery (that has to do work ) so that the current continues to flow.