M. Fabbrichesi , SISSA

3. Maxwell's equations in macroscopic media

This lecture is dedicated to Maxwell's equations in macroscopic media. Currents and charges are splitted into two terms ("external" and "induced") e.g: **J**=**J**_{ext}+**J**_{ind. }This lead to a rewriting of Maxwell's equation in terms of a new set of fields: **D**,**H**,**B**,**E **that keep into account the electric and magnetic answers of materials to applied fields [Jackson, Introduction I.4]. Concept of polarization and magnetization are introduced. It is seen that macroscopic fields such as **D **can be expanded in series containing dipole, quadrupole....terms [Jackson, Eq. I.9]. Next, boundary conditions at interfaces between different media are obtained by applying Maxwell's equations for the set **D**,**H**,**B**,**E **[Jack. Eqs. I.17-18 and I.19-20]. Then electrostatics and magnetostatics law are considered for introducing auxiliary fields called scalar and vector potentials that will be later very helpful for a Lagrangian description of electromagnetism. Finally, concerning the "reality" of the vector potential, the Aharonov-Bohm effect is briefly mentioned.