Page 379 - Electromagnetics

where G satisfies

2 ∂ 1 ∂

2
2
∇ − − G(r, t|r , t ) =−δ(r − r )δ(t − t ). (5.73)
2
2
v ∂t v ∂t 2
In § A.1 we find that
δ(t − t − R/v)

− (t−t )

G(r, t|r , t ) = e +
4π R

2 2
2 − (t−t ) I 1 (t − t ) − (R/v) R

+ e , t − t > ,
4πv (t − t ) − (R/v) 2 v
2
where R =|r − r |. For lossless media where σ = 0 this becomes

δ(t − t − R/v)

G(r, t|r , t ) =
4π R
and thus
∞ δ(t − t − R/v)

ψ(r, t) = S(r , t ) dt dV
V −∞ 4π R

S(r , t − R/v)

= dV . (5.74)
V 4π R
For lossless media, the scalar potentials and all rectangular components of the vector
potentials obey the same wave equation. Thus we have, for instance, the solutions to
(5.51):
i
µ J (r , t − R/v)

A e (r, t) = dV ,
4π V R
i
1 ρ (r , t − R/v)

φ e (r, t) = dV .
4π V R
These are called the retarded potentials since their values at time t are determined by the
values of the sources at an earlier (or retardation) time t − R/v. The retardation time is
determined by the propagation velocity v of the potential waves.
The fields are determined by the potentials:
i
i

1 ρ (r , t − R/v) ∂ µ J (r , t − R/v)

E(r, t) =−∇ dV − dV ,
4π V R ∂t 4π V R
i

1 J (r , t − R/v)

H(r, t) =∇ × dV .
4π V R
The derivatives may be brought inside the integrals, but some care must be taken when
the observation point r lies within the source region. In this case the integrals must be
performed in a principal value sense by excluding a small volume around the observation
point. We discuss this in more detail belowfor the frequency-domain fields. For details
regarding this procedure in the time domain the reader may see Hansen [81].
© 2001 by CRC Press LLC