Page 313 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 295

pleased with our definitions of V and A,
B =∇ × A (50)

∂V
∇ · A =−µ (54)
∂t
∂A
E =−∇V − (51)
∂t
The integral equivalents of (45) and (46) for the time-varying potentials follow
from the definitions (50), (51), and (54), but we shall merely present the final results
and indicate their general nature. In Chapter 11, we will find that any electromagnetic
disturbance will travel at a velocity
1
ν = √
µ
through any homogeneous medium described by µ and .In the case of free space,
8
this velocity turns out to be the velocity of light, approximately 3 × 10 m/s. It is
logical, then, to suspect that the potential at any point is due not to the value of the
charge density at some distant point at the same instant, but to its value at some
previous time, because the effect propagates at a finite velocity. Thus (45) becomes
[ρ ν ]

V = dν (57)
vol 4π R
where [ρ ν ] indicates that every t appearing in the expression for ρ ν has been replaced
by a retarded time,
R

t = t −
ν
Thus, if the charge density throughout space were given by
ρ ν = e −r cos ωt
then
R
[ρ ν ] = e −r cos ω t −
ν
where R is the distance between the differential element of charge being considered
and the point at which the potential is to be determined.
The retarded vector magnetic potential is given by

µ[J]

A = dν (58)
vol 4πR
The use of a retarded time has resulted in the time-varying potentials being given
thenameofretardedpotentials.InChapter14wewill apply(58)to thesimplesituation
of a differential current element in which I is a sinusoidal function of time. Other
simpleapplicationsof(58)areconsideredinseveralproblemsattheendofthischapter.