# 1 Maxwell’s Equations in Materials Electric field in dielectrics and conductors Magnetic field in materials Maxwell’s equations in its general form Boundary.

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1 Maxwell’s Equations in Materials Electric field in dielectrics and conductors Magnetic field in materials Maxwell’s equations in its general form Boundary conditions Full description of electromagnetic problems

2 Partition of Materials under Electric Field Materials are made of atoms with orbiting electrons and nuclei Under the external electric field, if electrons in atoms are not ionized to become free electrons – we have dielectrics Otherwise – we have conductors In some solids, electrons in atoms are ionized by the internal lattice (Coulomb) force without external field, such solids are certainly conductors

3 Polarization of Dielectrics Under the external electric field, electrons change their orbits to form dipoles (polarization), their volume density is defined as We then have

4 Electric Field in Dielectrics Under the external electric field, bound charges appear due to the polarization of the atoms; similar to free charges, bound charges excite electric field, hence Gaussian law in dielectrics should be modified to Where

5 Electric Field in Dielectrics We must be careful thatis a frequency domain expression. Only for “narrow band” electric field, the above expression is simplified to In time domain, we have

6 Electric Field in Conductors Inside the conductor, the most outside orbiting electrons in atoms are ionized to become free in the lattice of the host solid; under the external field, these electrons will move against the field As a result, the free electrons stay on one side of the conductor and leave ionized atoms on the opposite side; such a macroscopic separation of the electrons and ions will form a new field inside the conductor that cancels the external field Once the total field inside the conductor is not zero, the moving of electrons won’t stop; a steady state will exist only when the total field inside the conductor becomes zero

7 Electric Field in Conductors At steady state, the total internal electric field inside the conductor is zero At steady state, the electric potential is the same everywhere inside the conductor What about the external field is time varying? What about if the conductor has a limited (free) electron density, whereas the external field is so strong that no internal field can be further built up due to the exhaustion of (free) electrons?

8 Electric Field in Conductors What about if the conductor has a shape with complex topology?

9 Magnetization of Materials Similar to the introduction of the polarization under the electric field, we define the magnetization as We then have

10 Magnetic Field in Materials Under the external magnetic field, the magnetized current appears due to material magnetization; similar to the conventional conduction current, the magnetized current will excite the magnetic field; therefore, Ampere’s law is modified to Where

11 Summary on Static Electric-Magnetic Fields in Materials Field Description Potential Description Static Electric FieldStatic Magnetic Field

12 Electro-Magnetic Interaction in Materials Charge conservation law – the same Faraday’s law – the same Maxwell’s displacement current involves the Gaussian law for electric field, hence the Gaussian law for electric field in materials should be used accordingly

13 Maxwell’s Equations in its General Form 17 equations, yet 16 unknowns, one equation is redundant, e.g., the 4th equation is embedded in the 1st equation Why independent sources, i.e., the free charge and conduction current, have to be taken as unknowns as well?

14 Boundary Conditions Boundary conditions are self-contained in Maxwell’s equations Tangential field: Normal field: Scalar and vector potential (under Coulomb’s gauge): Conduction current linear density on boundary surface Free charge areal density on boundary surface Dielectric-dielectricDielectric-conductor or Uniform, dispersiveless

15 Inherent Field Divergence at Corner Points Boundary must be smooth, i.e., its 1st order derivative must be continuous Otherwise, electromagnetic field diverge at the 1st order derivative discontinued point

16 Field and Source Interaction Maxwell’s equations describe how electro-magnetic fields are excited by charge and current sources, and how they interact Electro-magnetic fields, in turn, act on sources, such action is aggregatively described by Lorentzian force Force density applied on charge density Force on charge

17 Filed and Source Interaction In solving the real-world problems, to avoid computational difficulties in statistical physics, the action of field on source is usually described by various phenomenological models established for different material systems A few typical examples –In vacuum (Newton’s law): –In conductors (Ohmic law): –In semiconductors (Drift-Diffusion model):

18 Full Description of Electromagnetic Problems Maxwell’s equations plus Lorentzian force Maxwell’s equations always take the same form Lorentzian force is usually replaced by problem dependent models, not in a unified form

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