Basic Theoretical Physics A Concise Overview

20.2 Retarded Scalar and Vector Potentials I: D’Alembert’s Equation 16520.2 Retarded Scalar and Vector Potentials I:D’Alembert’s EquationFor given electromagnetic fields E(r, t) and B(r, t) one can satisfy the secondand third Maxwell equations, i.e., Gauss’s magnetic law, divB = 0, andFaraday’s law of induction, curlE ≡ − ∂B , ∂twith the ansatz B(r, t) ≡ curlA(r, t) and E(r, t) ≡ −gradφ(r, t) − ∂A(r, t) . (20.4) ∂t The scalar potential φ(r, t) and vector potential A(r, t) must now becalculated simultaneously. However, they are not unique but can be “gauged”(i.e., changed according to a gauge transformation without any change of thefields) as follows: A(r, t) → A (r, t) := A (r, t) + gradf (r, t), (20.5) φ(r, t) → φ (r, t) := φ (r, t) − ∂f (r, t) . ∂tHere the gauge function f (r, t) in (20.5) is arbitrary (it must only bedifferentiable). (The proof that such gauge transformations neither changeE(r, t) nor B(r, t) is again based on the fact that differentiations can bepermuted, e.g., ∂ ∂f = ∂ ∂f .) ∂t ∂x ∂x ∂tIn the following we use this “gauge freedom” by choosing the so-calledLorentz gauge : divA + 1 ∂φ ≡ 0 . (20.6) c2 ∂tAfter a short calculation, see below, one obtains from the two remainingMaxwell equations (I and IV), divD = and curlH = j + ∂D , the so-called ∂td’Alembert-Poisson equations: − ∇2 − ∂2 φ(r, t) = E(r, t)/ε0 and c2∂t2 − ∇2 − ∂2 A(r, t) = μ0jB(r, t) . (20.7) c2∂t2c is the velocity of light in vacuo, while E and jB are the effective chargeand current density, respectively. These deviate from the true charge and truecurrent density by polarization contributions: E(r, t) := (r, t) − divP (rt) (20.8)and J (r, t) ∂P μ0 ∂t jB(r, t) := j(r, t) + + . (20.9)