Page 371 - Electromagnetics

Therefore the vector potential A e , which describes the solenoidal portion of both E and
B, is found from just the solenoidal portion of the current. On the other hand, the scalar
i
potential, which describes the lamellar portion of E, is found from ρ which arises from
i
∇· J , the lamellar portion of the current.
From the perspective of field computation, we see that the introduction of potential
functions has reoriented the solution process from dealing with two coupled first-order
partial differential equations (Maxwell’s equations), to two uncoupled second-order equa-
tions (the potential equations (5.24) and (5.28)). The decoupling of the equations is often
worth the added complexity of dealing with potentials, and, in fact, is the solution tech-
nique of choice in such areas as radiation and guided waves. It is worth pausing for
a moment to examine the form of these equations. We see that the scalar potential
obeys Poisson’s equation with the solution (5.25), while the vector potential obeys the
wave equation. As a wave, the vector potential must propagate away from the source
with finite velocity. However, the solution for the scalar potential (5.25) shows no such
behavior. In fact, any change to the charge distribution instantaneously permeates all
of space. This apparent violation of Einstein’s postulate shows that we must be careful
when interpreting the physical meaning of the potentials. Once the computations (5.17)
and (5.18) are undertaken, we find that both E and B behave as waves, and thus propa-
gate at finite velocity. Mathematically, the conundrum can be resolved by realizing that
individually the solenoidal and lamellar components of current must occupy all of space,
i
even if their sum, the actual current J , is localized [91].

The Lorentz gauge. A different choice of gauge condition can allowboth the vector
and scalar potentials to act as waves. In this case E may be written as a sum of two
terms: one purely solenoidal, and the other a superposition of lamellar and solenoidal
parts.
Let us examine the effect of choosing the Lorentz gauge
∂φ e
∇· A e =−µ − µσφ e . (5.29)
∂t
Substituting this expression into (5.26) we find that the gradient terms cancel, giving
2
2 ∂A e ∂ A e i
∇ A e − µσ − µ =−µJ . (5.30)
∂t ∂t 2
For lossless media
2
i
2
∇ A e − µ ∂ A e =−µJ , (5.31)
∂t 2
and (5.23) becomes
2 ρ i
2
∇ φ e − µ ∂ φ e =− . (5.32)
∂t 2
For lossy media we have obtained a second-order differential equation for A e , but φ e
must be found through the somewhat cumbersome relation (5.29). For lossless media
the coupled Maxwell equations have been decoupled into two second-order equations, one
involving A e and one involving φ e . Both (5.31) and (5.32) are wave equations, with J i
i
as the source for A e and ρ as the source for φ e . Thus the expected finite-velocity wave
nature of the electromagnetic fields is also manifested in each of the potential functions.
The drawback is that, even though we can still use (5.17) and (5.18), the expression for E
is no longer a decomposition into solenoidal and lamellar components. Nevertheless, the
choice of the Lorentz gauge is very popular in the study of radiated and guided waves.

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