Maxwell was the first to realize that Ampere’s law was incomplete as written in since there exist situations where ∇ ·J 6= 0 (especially the case for optical phenomena). Maxwell out that should be replaced with ∇·J = − ∂ρ ∂t (1.23) This is called the continuity equation for charge and current densities. Simply stated, if there is net current flowing into a volume there ought to be charge piling up inside. For the steady-state situation inherently considered by Ampere, the current into and out of a volume is balanced so that ∂ρ ± ∂t = 0.
Polarization of Materials
We are essentially finished with our analysis of Maxwell’s equations except for a brief discussion of current density J and charge density ρ. For convenience, it is common to decompose the current density into three categories: J = Free +Jam +Jpg (1.28) First, as you might expect, currents can arise from free charges in motion such as electrons in a metal, referred to as Free.
Second, individual atoms can exhibit internal currents that give rise to paramagnetic and diamagnetic effects, denoted by Jm. These are seldom important in optics problems, and so we will ignore these types of currents by writing Jam = 0. Third, molecules in a material can elongate, becoming dipoles in response to an applied electric field. This gives rise to a polarization current Jpg.
The polarization current is associated with a dipole distribution function P, called the polarization11 (in units of dipoles per volume, or charge times length per volume). Physically, if the dipoles change their strength or orientation as a function of time, an effective current density arises in the medium. Note that the time-derivative of an individual dipole moment renders charge times velocity.
Thus, the time derivative of ‘sloshing’ dipoles per volume gives a current density equal to
We seldom consider the propagation of electromagnetic waveforms through electrically charged materials, we will always write free = 0. One might be tempted in this case to assume that the overall charge density is zero, but this would be wrong. Even though the material is electrically neutral, the polarization P can vary in space, leading to local concentrations of positive or negative charges.
This type of charge density is denoted by up. It arises from no uniform arrangements of dipoles, as depicted in A polarized medium with ∇·P 6= 0. To connect ρp with P, we write the continuity equation (1.23) for the current and charge densities associated with the polarization:
The Wave Equation
Hen Maxwell unified electromagnetic theory, he immediately noticed that waves are solutions to his set of equations. In fact his desire to find a set of equations that allowed for waves aided his effort to find the correct equations. After all, it was already known that light traveled as waves. Kirchhoff had previously pointed out that. The may give the impression that you could always just draw a surface that avoids cutting any dipoles.
However, the function P(r) is continuous, while the figures depict crudely just a few dipoles. In a continuous material, you can’t draw a surface that avoids cutting dipoles. 13It is not uncommon to see the macroscopic Maxwell equations written in terms of two auxiliary fields: H and D. The field H is useful in magnetic materials.
In these materials, the combination B ± µ0 in Ampere’s law is replaced by H ≡ B/µ0 −M, where Jam = ∇ ×M is the current associated with the material’s magnetization.
Since we only consider nonmagnetic materials (M = 0), there is little point in using H. The field D, called the displacement, is defined as D ≡ ²0E+P. This combination of E and P occurs in Coulomb’s law and Ampere’s law. For physical clarity, the authors of this book elect to retain the prominence of the polarization P in the equations.