Download Maxwell`s equation

Maxwell Was Right
A posting was made to the Thunderbolts Forum on the Future of Science by a person who quotes
an article criticizing the work of James Clark Maxwell. On page 8/23 of this paper the author
says:
“Ampere’s original law allows the calculation of the magnetic field B produced at a
point in space by currents J flowing along other curves in space. It has its experimental
roots in Oersted’s great discovery that an electric current produces a magnetic field in the
space around it. If another term is added to this equation, it follows that the magnetic
field can be produced also in the manner described by this new term. Adding Ampere’s
original equation is equivalent to saying that a changing electric field E can produce a
magnetic field B . Maxwell’s modification of Ampere’s law by the addition of this
supplementary term is not correct.”
This author is mistaken. Maxwell is absolutely correct. Maxwell’s modification of Ampere’s law
by the addition of this supplementary term is to allow for the current through a capacitor (known
as a ‘displacement current’) to be taken into account. A basic idea in circuit theory is that a timevarying voltage across a capacitor will produce a current through (into the positive terminal and
out of the negative terminal of) that capacitor. Positive charge accumulates on one plate and an
equal amount is taken away from the other plate. Doing this results in a terminal current, I.
Assuming the insulation (dielectric) inside the capacitor, between the plates, is of high quality,
there is no flow of charge inside the capacitor from the + plate to the – plate. But a magnetic
field is never-the-less produced in that region as if there were a current, I, there.
Maxwell accounts for this as follows: His Curl of B equation can be expressed as
E
.
(1)
  B   0 J   0 0
t
This is written a bit more efficiently (by using B   0 H ) as
E
.
(2)
  H  J  0
t
This is a ‘point equation’ so it deals with current-densities rather than just currents. All three
terms in (2) have units of current-density (Amps/m2). The terms on the right therefore signify
two current-density paths in parallel (they add). The J term can be thought of as coming from a
resistor of length, l, area, Ar, and resistivity, ρ. This path would occur if a capacitor was ‘leaky’
(had a dielectric that allowed a non-zero valued current to go through the dielectric in response
to the applied voltage, V). The units of ρ are Ohm-meters because the resistance, R, of a resistor
in Ohms is given by R = ρl/ Ar or in units, Ohms = (Ohm-m)(m)/m2.
Now for the last term: Consider a capacitor that has a perfect dielectric such that J = 0. What
would be occurring at a point, p, in between the plates? The definition of a capacitor in terms of
its circuit variables (terminal current and voltage) is:
IC  C
V
dV
or, if the capacitor has a fixed shape, size, and value, I C  C
.
dt
t
(3)
If the plates are parallel, C 
0 A
, where ε0 is the permittivity of the dielectric between the
d
plates, and d is the separation between them. With E = V/d, and using (3) we can rewrite (2) as
 
V A
E
d  J  C V 1
 J  0
(4)
t
A t
t A
All terms in expression (4) are current density (Amps/m2). And the last term in (4) is called the
‘displacement current’. So in a perfectly insulated capacitor, even with very widely spaced
plates, a magnetic field will still be produced at points such as, p, even though no ohmic current
passes through them. Displacement current will only exist if the capacitor has a time-varying
voltage across it, and is just as proficient at creating a magnetic field as an equal valued terminal
current.
  H  J  0
 
Figure 1. The “displacement current density” that exists at point, p, is equal in magnitude to I/A thus
preserving the continuity of I (and thus the B-field) throughout the circuit. Ohmic current, J, does not exist in
the gap between the capacitor plates.
The cylindrical magnetic field produced by current, I, is continuous (surrounding the capacitor as
well as the conductors carrying I) even though no conductive current density exists between the
plates. Ideally, there is no lapse in the cylindrically shaped B-field no matter how widely
separated the plates are. So, in cosmic space, if there is a build-up of positive charge in one
region and a build-up of negative charge in another region, the time-varying voltage difference
between those two regions will create a B-field around a path between those two regions just as
if a current were flowing in that path – even if no ohmic path exists.
Understanding the ramifications of displacement current is difficult for many students, but that is
no reason for claiming a falsification of one of Maxwell’s crowning achievements.
D.E. Scott