Lorenz Gauge and Inhomogeneous Wave Equation PDF

3 Lorenz gauge and inhomogeneous wave equa- z ρ(r′ ) ′...

3 Lorenz gauge and inhomogeneous wave equa- z ρ(r′ ) ′ r−r tion J(r′) r r′ Last lecture we found out that given the static sources ρ = ρ(r) and J = J(r), y O static fields x E = −∇V and B = ∇ × A satisfying ✬ ✬ ✩ ✩ Electrostatics: (curl-free) Magnetostatics: (divergence-free) ∇·D = ρ ∇·B = 0 ∇×E = 0 ∇×H = J ✫ D = ϵo E ✪ ✫ B = µo H ✪ can be computed using the potentials ✬ ✩ ✬ ✩ ρ(r ) d r, µoJ(r′ ) 3 ′ ′ A(r) = ! V (r) = d3 r′ , ! ′ 4π|r − r′| 4πϵo|r − r | the solution of the solution of ρ ✫ ∇2A = −µoJ. ✪ ∇2 V = −. ✫ ϵo ✪ 1  Over the next two lectures we will explain why in case of time-varying sources ρ = ρ(r, t) and J = J(r, t), ∇·D = ρ the full set of Maxwell’s equations (see margin) can be satisfied by ∇·B = 0 ∂A ∂B ∂t ∇×E = − ∂t E = −∇V − and B = ∇ × A ∂D in terms of delayed or retarded potentials specified as ∂t ∇×H = J+ ✬ ✩ ✬ ✩ ′ ′ | | c d r, µoJ(r′, t − |r−r d r, A(r, t) = c ) 3 ′ ρ(r′ , t − |r−r V (r, t) = ) 3 ′ ! ! 4πϵo|r − r′| 4π|r − r′ | the solution of inhomogeneous wave equation the solution of inhomogeneous wave equation 2 ∂ 2V ρ ∂ 2A ∂t2 ∇ 2 A − µ o ϵo = −µoJ ∂t ∇ V − µ o ϵo 2 = − ✫ ϵo ✫ ✪ ✪ 1 z ρ(r′ , t) is the speed of light in free space. r − r′ µ o ϵo where c ≡ √ ′ J(r′ , t) Note that retarded potentials r r V (r, t) and A(r, t) y are essentially weighted and delayed sums of charge and current densi- O ties x 2  ρ(r, t) and J(r, t), while the fields E and B are obtained by spatial and temporal deriva- z ρ(r′ , t) tives of the potentials. r − r′ ′ J(r′ , t) Alternatively, we can first use r r ′ | c y µoJ(r′, t − |r−r A(r, t) = ) 3 ′ ! d r ⇒ B=∇×A 4π|r − r′ | O and then find the anti-derivative of Ampere’s law x J⇒A⇒B⇒E B ∂E = ϵo ∂t ∇× µo to determine E outside the region where J is non-zero, bypassing the use of scalar retarded potential V (r, t) — that is the most common approach used in radiation studies. We will next verify the procedure outlined above and the start discussing its applications in radiation studies. 3  The full set of Maxwell’s equations is repeated in the margin for con- venience. Divergence-free nature of B compels us to define a vector ∇·D = ρ potential A via ∇·B = 0 ∂B ∂t B=∇×A ∇×E = − just as before. Inserting this in Faraday’s law we get ∂D ∂t ∇×H = J+ ∂ ∂A ) = 0. ∂t ∂t ∇×E=− ∇×A ⇒ ∇ × (E + Evidently ∂A ∂A E+ is curl free, so it must be true that E + ∂t ∂t = −∇V, or ∂A ∂t E = −∇V − in terms of some scalar potential V. Main difference from statics appears to be the need for two poten- tials, instead of one, to represent the electric field E under time- varying conditions. We continue.... Now substitute ∂A ∂t B = ∇ × A and E = −∇V − 4  in the remaining two Maxwell’s equations — Gauss’s and Ampere’s laws ∂ o B) = J + (ϵoE), ∂t ∇ · (ϵoE) = ρ and ∇ × (µ−1 that we have not touched yet. Upon substitutions we get ∂A ∂ ∂A ), ∂t ∂t ∂t ϵo∇ · (−∇V − ) = ρ and ∇ " × #$ ∇ × A% = µoJ + µoϵo (−∇V − ∇(∇ · A) − ∇2A which looks like a big mess. But if we specify ∂V (Lorenz gauge) ∂t ∇ · A = −µoϵo these messy equations simplify as ∂ 2V ρ 2 ∂ 2A = and A µ o ϵ o ∂t2 ∂t2 ∇ 2 V − µ o ϵo − ∇ − = −µoJ ϵo which we recognize as the inhomogeneous or “forced” wave equations for V and A stated earlier on. – The derivation of the decoupled wave equations above hinged upon our use of Lorenz gauge which reduces to the Coulomb gauge, ∇ · A = 0, in static situations. – Note also that the forced wave equations reduce to Poisson’s equa- tions under time-static conditions. 5  – Since we know how to solve the unforced wave equation from ECE 329, and since we know how to solve the Poisson’s equation, it is now a matter of combining those methods to solve the forced wave equations obtained above. Just a few additional comments on gauge selection before we go on (next lecture): Gauge selection amounts to deciding what to assign to ∇ · A. We can make any assignment that pleases us. This is like choosing the ground node in a circuit problem. Whatever simplifies the problem the most is the best gauge to use. – Lorenz gauge is clearly a good one since it led to decoupled wave equations which are very convenient to work with. We can attack the decoupled equations for V and A one at a time. 6