Page 312 - Engineering Electromagnetics, 8th Edition

294 ENGINEERING ELECTROMAGNETICS

and
∂
2 ρ ν (53)
∇ V + (∇ · A) =−
∂t
There is no apparent inconsistency in (52) and (53). Under static or dc conditions
∇ · A = 0, and (52) and (53) reduce to (48) and (47), respectively. We will therefore
assume that the time-varying potentials may be defined in such a way that B and
E may be obtained from them through (50) and (51). These latter two equations do
not serve, however, to define A and V completely. They represent necessary, but not
sufficient, conditions. Our initial assumption was merely that B =∇ × A, and a
vector cannot be defined by giving its curl alone. Suppose, for example, that we have
avery simple vector potential field in which A y and A z are zero. Expansion of (50)
leads to
B x = 0
∂A x
B y =
∂z
∂A x
B z =−
∂y
and we see that no information is available about the manner in which A x varies with
x. This information could be found if we also knew the value of the divergence of A,
for in our example
∂ A x
∇ · A =
∂x
Finally, we should note that our information about A is given only as partial derivatives
and that a space-constant term might be added. In all physical problems in which the
region of the solution extends to infinity, this constant term must be zero, for there
can be no fields at infinity.
Generalizing from this simple example, we may say that a vector field is defined
completely when both its curl and divergence are given and when its value is known at
any one point (including infinity). We are therefore at liberty to specify the divergence
of A, and we do so with an eye on (52) and (53), seeking the simplest expressions.
We define
∂V
∇ · A =−µ (54)
∂t
and (52) and (53) become
2
∂ A
2
∇ A =−µJ + µ (55)
∂t 2
and
2
∂ V
2 ρ ν (56)
∇ V =− + µ 2
∂t
These equations are related to the wave equation, which will be discussed in
Chapters 10 and 11. They show considerable symmetry, and we should be highly