Chapter 8: Magnetic Vector Potential

Illustration 3

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.

Notes: The figures may pop up in a separate window as the user clicks on an item.

 
Figures (click to enlarge)
Figure 1

Figure 2a

Figure 2b

Figure 1: The vector potential in the cross-section of a wire with uniform current distribution.

 

Figure 2: Comparison between the magnetic vector potential component  of a wire with uniformly distributed current and the electric potential V of the equivalent cylinder with uniformly distributed charge.