Until the 19th century electric and magnetic effects were seen as two independent physical occurences. In 1820 Oersted proved, that electric currents can influence the needle of a compass. Ampère and Faraday laid the foundation for the unified theory of electrodynamics, which was elaborated by James Clerk Maxwell (1831-1879).
His famous equations are the starting point for our investigations:
(
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In order to solve these equations for our purpose, we have to make several assumptions. First, we neglect any displacement currents , which are typically relevant only at radio frequencies (in the MHz regime), and set
. In the quasistatic approximation we omit the the term
(
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Secondly, we assume, that there are no free charges (. The simplified Maxwell equations are given by
(
)
Then, we require the field intensity and the flux intensity
to obey the constitutive relationship
. (
)
If the material is nonlinear (e.g. ferromagnetic), the permeability ( is a function of B.
(
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However, we will consider only linear materials in this case in this project.
The relationship between the electric field intensity and the current density
is given by
. (
)
Now we introduce a magnetic vector potential , and define it as
(
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which guarantees the validity of Maxwell's second equation. Then we can rewrite the fourth equation and obtain
(
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With the Coulomb gauge condition
(
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and the well known operator relation
(
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we find
. (
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Inserting (7) in the third Maxwell equation yields
. (
)
In the case of 2D problems, we can integrate this equation and get
(
)
and together with constitutive relationship between the electric field intensity and the current density
we obtain
(
)
Finally we insert this in (11) to eliminate and arrive at
, (
)
where represents the applied current sources.
Finally, we restrict ourselves to time harmonic problems, in which all fields oscillate harmonically at one fixed frequency. Thus, we can use a phasor transformation and rewrite the magnetic vector potential as
(
)
in which is the complex amplitude. By substituting this ansatz in (15) we can eliminate the time derivative and finally arrive at
. (
)
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