old Appendix A 4 and 5 REVIEWED

Old App A facts 4 and 5 (replaced by a single fact 4) PhL 9.18.13 This is just backup storage, will be thrown out eventually. 4. Fact 4: If div B = 0 and curl E = - ∂B/∂t , then there exist both A and φ such that B = curl A and E = - grad φ-∂A/∂t, and the quantity div A may be set to any function f. Proof: We know from Fact 2 that A exists such that B = curl A and such that div A equals any arbitrary function f (we dispense with prime on A). Consider the vector field E' = E + ∂A/∂t. Apply curl to find that curl E' = curl E + ∂B/∂t = 0. Now apply Fact 3 to this vector E' to conclude that there exists φ such that E' = -grad φ. But this says E = -grad φ - ∂A/∂t. Thus we have expressly found A and φ which satisfy the requirements of Fact 4. 5. Fact 5: Assume that potentials A and φ have been obtained as in Fact 4. There exists an infinite set of other potentials A' and φ' which yield the same E and B fields. They are given by A' = A + grad Λ dim(Λ) = amp-henry = volt-sec φ' = φ - ∂Λ/∂t (A.5.1) where Λ is an arbitrary scalar function. It is by making a particular such transformations that div A' can be set to an arbitrary selected function f, as discussed in Fact 2. Proof: We already showed in Fact 2 that A' = A + grad Λ does not alter B. The second condition of (A.5.1) is required so that E = -grad φ - ∂A/∂t also remains unaltered. That is, E' = - grad φ'- ∂A' /∂t = -grad (φ - ∂Λ/∂t) - ∂/∂t (A + grad Λ) = - grad φ - ∂A/∂t = E . The gauge transformation (A.5.1) appears in Jackson as (6.12) and (6.13).