The magnetic vector
potential A (Wb/m) is usually defined
through its relation to the magnetic flux density B as
, T.
Since (Ampère’s
law), and , a partial differential
equation is derived for A as
.
This is an important
equation in magnetostatics—it is the basis
of all numerical algorithms used in related computer-aided
design.
In a homogeneous
medium, the above equation becomes the vector
Poisson equation
.
In rectangular coordinates,
this equation conveniently reduces to three independent
scalar Poisson equations:
.
Remember that the
Poisson equation governs the electric potential V in
electrostatics, too: . Thus, a number of
magnetostatic problems are in fact very similar
to their electrostatic counterparts.
It is also important
to note that in a uniform medium the direction
of A is
the same as the direction of the current. If
the current is along the z-axis, for example,
then A has only a z-component, .
Figure
1 shows a vector plot of A in
the x-y plane. Here, A is
due to a current (total current is A), which is
uniformly distributed in the cross-section
of a rather thick wire of radius mm. The wire
is along the z-axis, and, thus, the
current density has only a z-component, . That is why we see
only the tips of the arrows depicting .
To compare the magnetic
potential to the electric
potential V, we also compute V due
to the same thick cylinder only that this time
it is uniformly charged with C/m3.
The two plots in Figure
2 confirm the same mathematical behavior
of both potential functions.
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