Page 228 - Engineering Electromagnetics, 8th Edition

210 ENGINEERING ELECTROMAGNETICS

7.6 THE SCALAR AND VECTOR
MAGNETIC POTENTIALS
The solution of electrostatic field problems is greatly simplified by the use of the
scalar electrostatic potential V. Although this potential possesses a very real physical
significance for us, it is mathematically no more than a stepping-stone which allows
us to solve a problem by several smaller steps. Given a charge configuration, we may
first find the potential and then from it the electric field intensity.
We should question whether or not such assistance is available in magnetic fields.
Can we define a potential function which may be found from the current distribution
and from which the magnetic fields may be easily determined? Can a scalar magnetic
potential be defined, similar to the scalar electrostatic potential? We will show in
the next few pages that the answer to the first question is yes, but the second must
be answered “sometimes.” Let us attack the second question first by assuming the
existence of a scalar magnetic potential, which we designate V m , whose negative
gradient gives the magnetic field intensity

H =−∇V m

The selection of the negative gradient provides a closer analogy to the electric potential
and to problems which we have already solved.
This definition must not conflict with our previous results for the magnetic field,
and therefore

∇× H = J =∇ × (−∇V m )

However, the curl of the gradient of any scalar is identically zero, a vector identity
the proof of which is left for a leisure moment. Therefore, we see that if H is to be
defined as the gradient of a scalar magnetic potential, then current density must be
zero throughout the region in which the scalar magnetic potential is so defined. We
then have

H =−∇V m (J = 0) (43)

Because many magnetic problems involve geometries in which the current-carrying
conductors occupy a relatively small fraction of the total region of interest, it is evident
that a scalar magnetic potential can be useful. The scalar magnetic potential is also
applicable in the case of permanent magnets. The dimensions of V m are obviously
amperes.
This scalar potential also satisfies Laplace’s equation. In free space,

∇ · B = µ 0 ∇ · H = 0

and hence

µ 0 ∇ · (−∇V m ) = 0