into Ampere's and Faradays laws. It is most convenient to analyze the relationship Ising superposition of the cases for which Jm =0 and J=0. With Jm=0 Faraday's law V×E (5.16) Since v xe is solenoidal that B=0, which is equivalent to the auxiliary Maxwell equation V.B= 0. Now, substitution of (5. 14)and(5.15)into(5.16)gives Using V×(VφE)=0 and combining the terms we get da V×v×AE+ dAg + ve Substitution into(. 14)gives +VφE+V] Combining the two gradient functions together, we see that we can write both e and B in terms of two potentials dA E= Vφe, B=V×A (518) where the negative sign on the gradient term is introduced by convention. Gauge transformations and the Coulomb gauge. We pay a price for the simplicity of using only two potentials to represent E and B. While V x ae is definitely solenoidal, Ae itself may not be: because of this (5.17) may not be a decomposition into solenoidal and lamellar components. However, a corollary of the Helmholtz theorem states that a vector field is uniquely specified only when both its curl and divergence are specified. Here there is an ambiguity in the representation of E and B; we may remove this ambiguity and define Ae uniquely by requiring that Then Ae is solenoidal and the decomposition (5.17) is solenoidal-lamellar. This require- ment on Ae is called the Coulomb gauge. The ambiguity implied by the non-uniqueness of V. Ae can also be expressed by the observation that a transformation of the type A→A+Vr, (5.20) (5.21) ②2001into Ampere’s and Faraday’s laws. It is most convenient to analyze the relationships using superposition of the cases for which Jm = 0 and J = 0. With Jm = 0 Faraday’s lawis ∇ × E = −∂B ∂t . (5.16) Since ∇ × E is solenoidal, B must be solenoidal and thus ∇φB = 0. This implies that φB = 0, which is equivalent to the auxiliary Maxwell equation ∇ · B = 0. Now , substitution of (5.14) and (5.15) into (5.16) gives ∇ × [∇ × AE + ∇φE ] = − ∂ ∂t [∇ × AB] . Using ∇ × (∇φE ) = 0 and combining the terms we get ∇ ×  ∇ × AE + ∂AB ∂t  = 0, hence ∇ × AE = −∂AB ∂t + ∇ξ. Substitution into (5.14) gives E = −∂AB ∂t + [∇φE + ∇ξ ] . Combining the two gradient functions together, we see that we can write both E and B in terms of two potentials: E = −∂Ae ∂t − ∇φe, (5.17) B =∇× Ae, (5.18) where the negative sign on the gradient term is introduced by convention. Gauge transformations and the Coulomb gauge. We pay a price for the simplicity of using only two potentials to represent E and B. While ∇ × Ae is definitely solenoidal, Ae itself may not be: because of this (5.17) may not be a decomposition into solenoidal and lamellar components. However, a corollary of the Helmholtz theorem states that a vector field is uniquely specified only when both its curl and divergence are specified. Here there is an ambiguity in the representation of E and B; we may remove this ambiguity and define Ae uniquely by requiring that ∇ · Ae = 0. (5.19) Then Ae is solenoidal and the decomposition (5.17) is solenoidal–lamellar. This require￾ment on Ae is called the Coulomb gauge. The ambiguity implied by the non-uniqueness of ∇ · Ae can also be expressed by the observation that a transformation of the type Ae → Ae + ∇, (5.20) φe → φe − ∂ ∂t , (5.21)