Maxwell's Equations

Maxwell's Equation in Free Space

Taking divergence of equation (iii) we get

 So equation (iii) is valid.
Again taking divergence of equation (iv) we have  
the left side must be zero, but the right side, in general, is not.  
For steady currents  but for time varying fields   the Ampere's law can not be right.

How Maxwell fixed Ampere's Law

From continuity equation and Gauss Law

Thus

A changing electric field induces a magnetic field.
Maxwell called this extra term the displacement current

Integral form of Ampere's law

Paradox of Charging Capacitor

Displacement current resolves the paradox of charging capacitor. If the capacitor plates are very close together, then the electric field between them is:   where Q is the charge on the plate and A is its area.

Thus, between the plates

If we choose the flat surface, then E = 0 and I_enc =I

If, on the other hand, we use the balloon-shaped surface, then I_enc = 0 , then

So we get the same answer for either surface, though in the first case it comes from the genuine current and in the second from the displacement current.

Maxwell's Equation in Free Space

Maxwell's Equation in Linear Isotropic Media
For materials that are subject to electric and magnetic polarization there is more convenient way to write Maxwell's equations. Inside polarized matter there will be accumulation of "bound" charge and current over which we don't have direct control. So we will reformulate Maxwell's equation in such a way as to make explicit reference only to those sources we control directly: the "free" charges and currents.

We know that an electric polarization  produces a bound charge density  
Likewise, a magnetic polarization (or "magnetization")  results in a bound current  

Due to time varying field any change in the electric polarization involves a flow of (bound) charge, (call it polarization current  ) which must be included in the total current.

Consider a small piece of polarized material. The polarization introduces a charge density σ_b = P at one end and -σ_b at the other  If P now increases a bit, the charge on each end increases accordingly, giving a net current-carrying,

Thus the polarization current  

The po larizat ion current has nothing to do with the bound current,  . The bound current  is associated with magnetization of the material and involves the spin and orbital motion of electrons.

In view of all this, the total charge density
and the total current density
Gauss's law can now be written as:

Now, Ampere's law (with Maxwell's term) becomes

In terms of free charges and currents, then, Maxwell's equations read

(i)
(ii)

(iii)

(iv)
For linear media,

and displacement current

Integral form

Boundary Conditions on the Fields at Interfaces

In particular, if there is no free charge or free current at the interface between medium1 and medium 2, then