Therefore the vector potential Ae, 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 potential, which describes the lamellar portion of E, is found from p' which arises from V- 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 aw 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 Einsteins postulate shows that we must be careful hen 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 even if their sum, the actual current J, is localized [911 The Lorentz gauge. a different choice of gauge condition can allow both 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 Let us examine the effect of choosing the lorentz gauge (5.29) Substituting this expression into(5.26)we find that the gradient terms cancel, giving (5.30) a2A (531) and(5.23)become For lossy media we have obtained a second-order differential equation for Ae, but e must be found through the somewhat cumbersome relation(5.29). For lossless media e coupled Maxwell equations have been decoupled into two second-order equations, one involving Ae and one involving e. Both(5.31)and(5.32) are wave equations, with J as the source for Ae and p' as the source for 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 ②2001Therefore the vector potential Ae, 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 potential, which describes the lamellar portion of E, is found from ρi which arises from ∇ · Ji , 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, even if their sum, the actual current Ji , 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 ∇ · Ae = −µ ∂φe ∂t − µσφe. (5.29) Substituting this expression into (5.26) we find that the gradient terms cancel, giving ∇2 Ae − µσ ∂Ae ∂t − µ ∂2Ae ∂t 2 = −µJi . (5.30) For lossless media ∇2 Ae − µ ∂2Ae ∂t 2 = −µJi , (5.31) and (5.23) becomes ∇2 φe − µ ∂2φe ∂t 2 = −ρi . (5.32) For lossy media we have obtained a second-order differential equation for Ae, but φe 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 Ae and one involving φe. Both (5.31) and (5.32) are wave equations, with Ji as the source for Ae and ρi as the source for φe. 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.