At about the time of the American Civil War, James Clerk Maxwell made an attempt
to combine the best mathematics of his day with all of the experimental work
on electricity and magnetism from the preceding hundred years.
As he did so, he was mystified by Faraday's idea that the stored energy
in a capacitor was stored in the electric field between the plates.
Was the energy density formula,
just a formula, or was the energy somehow really stored in space? As he thought about this formula, he realized that in a dielectric it was possible to see how the energy could be stored: it was stored in the stretching of the atoms of the material. The larger the electric field, the more the atoms were stretched, and when the electric field was removed, the atoms snapped back to their original state, giving up the energy that was stored in them. Taking this as a hint, Maxwell made the hypothesis that the vacuum was not really empty at all, but was instead filled with atoms of a very fine and insensible material which he called the ether. When electric energy was stored in space, Maxwell took this to mean that the atoms of the ether became stretched, just like the atoms in paper or oil.
Once he came to believe in this picture, he was led to the following
brilliant insight: if ether atoms become stretched when an electric field is
applied, then when the electric field is changing in time, there
must be a current in the ether.
This must be so because when an atom becomes more stretched by
the increasing electric field, its positive charge
moves in the direction of the applied electric
field while its negative charge moves in the opposite direction.
But this means that both moving charges contribute to current flow in the direction
of the applied electric field.
This current, Maxwell realized, must produce magnetic field, and hence should
be added to the conduction current in Ampere's law.
He called this current displacement current and found the following
formula for the displacement current flowing through a surface:
where 
is our old friend, the electric flux through the surface.
This formula can be used to find the total amount of displacement
current flowing through a surface, but it doesn't indicate its direction.
The direction is given by the following formula
for the displacement current density 
:
As with conduction current, the displacement current density is the displacement current
per unit area.
Because of the time derivative, an increasing electric field makes displacement current
in the direction of 
, while a decreasing electric field makes displacement
current in the direction opposite to .
The physical meaning of this displacement current is that a changing electric field makes
a changing magnetic field.
To find the direction of the magnetic field produced by this effect
we use our usual right-hand rule
for currents: point your thumb in the direction of the displacement current
(the direction of 
), and your fingers will curl around in the direction
of the magnetic field.
(Note that if the electric field is decreasing in time, then the negative time
derivative gives a displacement vector that points opposite to .)