Student Nechay V., Ph.D. Kovalenko M.

National Technical University of Ukraine «Kyiv Polytechnic Institute», Ukraine

CALCULATION OF THE ELECTROMAGNETIC ENERGY

Electromagnetic energy is a form of energy contained in an electromagnetic field. Also included are special cases of a pure electric field and the net magnetic field. In many cases, the electric and magnetic energy are closely related to each other, each of them can be considered as "downside" of another one. This energy is the mechanical work done by moving charges and conductors in electrical and magnetic fields.

The electric and magnetic energies are defined as [1]:

(1)

(2)

Electrical energy is one of the most important form of energy. Electricity in its final form can be transmitted over a long distances to the consumer.

The time derivatives of these expressions are the electric and magnetic power [1]:

(3)

(4)
These quantities are related to the resistive and radiative energy, or energy losses, through Poynting’s theorem:

(5)

where V is the computation domain and S is the closed boundary of V. Poynting theorem - a theorem describing the law of an electromagnetic field energy conservation . The theorem was proved in 1884 by John Henry Poynting. It comes down to the following formula [2]:

(6)

The first term on the right-hand side represents the resistive losses:
(7)

which result in heat dissipation in the material. (The current density J in this
expression is the one appearing in Maxwell-Ampère’s law.)
The second term on the right-hand side of Poynting’s theorem represents the radiactive losses [2]:

(8)

The quantity is called the Poynting vector.

Under the assumption the material is linear and isotropic, it holds that:

(9)

(10)

By interchanging the order of differentiation and integration (justified by the fact that the volume is constant and the assumption that the fields are continuous in time), the result is [1]:

(11)

The integrand of the left-hand side is the total electromagnetic energy density:
(12)

For electric and magnetic fields energy is proportional to the square of the field strength. Strictly speaking, the term "electromagnetic energy" is not quite correct.

The calculation of the total energy of the electric field, even one electron leads to a value equal to infinity, as the corresponding integral diverges. The infinite energy of the field is quite the end of the electron is one of the theoretical problems of classical electrodynamics. Instead, the physics is usually used the concept of the energy density of the electromagnetic field (to a certain point in space). The total energy of the field is equal to the integral of the energy density over the entire space. The energy density of the electromagnetic field is the sum of the energy densities of electric and magnetic fields.

References:

1. J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons, NewYork,1993.
2. B.D. Popovic, Introductory Engineering Electromagnetics, Addison-Wesley,
Reading, Massachusetts, 1971.