Current and Charges as Sources of Fields

So far we discussed the propagation of electromagnetic waves without external sources (i.e. when pe = 0 and Jg = 0). To solve the Maxwell s equations in presence of sources, we can introduce the magnetic vector potential A and the electric scalar potential (p so that  

The magnetic vector potential always exists as H is divergence-less (see Eq. (1.2), for non magnetic media) and thus it can be written as Eq. (1.86). The electric scalar potential also always exists in fact from Eq. (1.3) we obtain  

The magnetic vector potential and the electric scalar potential are not uniquely defined. A particular form of the A and rj) is called gauge. Here we will consider the Lorentzgauge which is defined by 

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