Page 232 - Engineering Electromagnetics, 8th Edition

214 ENGINEERING ELECTROMAGNETICS

Equation (46) serves as a useful definition of the vector magnetic potential A.
Because the curl operation implies differentiation with respect to a length, the units
of A are webers per meter.
As yet we have seen only that the definition for A does not conflict with any
previous results. It still remains to show that this particular definition can help us to
determine magnetic fields more easily. We certainly cannot identify A with any easily
measured quantity or history-making experiment.
We will show in Section 7.7 that, given the Biot-Savart law, the definition of B,
and the definition of A, A may be determined from the differential current elements by

µ 0 IdL

A = (47)
4πR
The significance of the terms in (47) is the same as in the Biot-Savart law; a direct
current I flows along a filamentary conductor of which any differential length dL is
distant R from the point at which A is to be found. Because we have defined A only
through specification of its curl, it is possible to add the gradient of any scalar field
to (47) without changing B or H, for the curl of the gradient is identically zero. In
steady magnetic fields, it is customary to set this possible added term equal to zero.
The fact that A is a vector magnetic potential is more apparent when (47) is
compared with the similar expression for the electrostatic potential,
ρ L dL

V =
4π 0 R
Each expression is the integral along a line source, in one case line charge and in the
other case line current; each integrand is inversely proportional to the distance from
the source to the point of interest; and each involves a characteristic of the medium
(here free space), the permeability or the permittivity.
Equation (47) may be written in differential form,
µ 0 IdL
dA = (48)
4π R
if we again agree not to attribute any physical significance to any magnetic fields we
obtain from (48) until the entire closed path in which the current flows is considered.
With this reservation, let us go right ahead and consider the vector magnetic
potential field about a differential filament. We locate the filament at the origin in free
space, as shown in Figure 7.19, and allow it to extend in the positive z direction so
that dL = dz a z .We use cylindrical coordinates to find dA at the point (ρ, φ, z):

µ 0 Idz a z
dA =
2
4π ρ + z 2
or
µ 0 Idz
dA z = dA φ = 0 dA ρ = 0 (49)
2
4π ρ + z 2