Page 311 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 293

the curl that is identically zero. Let us therefore tentatively accept (50) as satisfactory
for time-varying fields and turn our attention to (49).
The inadequacy of (49) is obvious because application of the curl operation to
each side and recognition of the curl of the gradient as being identically zero confront
us with ∇× E = 0. However, the point form of Faraday’s law states that ∇× E is
not generally zero, so let us try to effect an improvement by adding an unknown term
to (49),
E =−∇V + N
taking the curl,
∇× E = 0 +∇ × N
using the point form of Faraday’s law,
∂B
∇× N =−
∂t
and using (50), giving us
∂
∇× N =− (∇× A)
∂t
or
∂A
∇× N =−∇ ×
∂t
The simplest solution of this equation is
∂A
N =−
∂t
and this leads to
∂A
E =−∇V − (51)
∂t
We still must check (50) and (51) by substituting them into the remaining two of
Maxwell’s equations:
∂D
∇× H = J +
∂t
∇ · D = ρ ν
Doing this, we obtain the more complicated expressions
2
1 ∂V ∂ A
∇× ∇× A = J + −∇ −
µ ∂t ∂t 2
and
∂

−∇ · ∇V − ∇ · A = ρ ν
∂t
or
2
∂V ∂ A
2
∇(∇ · A) −∇ A = µJ − µ ∇ + (52)
∂t ∂t 2