Embedded in the laws of Gauss and Ampère is a relationship that
must exist between the charge and current densities. To see this,
first apply Ampère's law to a closed surface, such as sketched in
Fig. 1.5.1. If the contour C is regarded as the"drawstring" and
S as the "bag," then this limit is one in which the "string" is
drawn tight so that the contour shrinks to zero. Thus, the open
surface integrals of (1.4.1) become closed, while the contour integral
vanishes.
But now, in view of Gauss' law, the surface integral of the electric
displacement can be replaced by the total charge enclosed. That is,
(1.3.1) is used to write (1) as
This is the law of conservation of charge. If there is a net current
out of the volume shown in Fig. 1.5.2, (2) requires that the net
charge enclosed be decreasing with time.
Figure 1.5.1. Contour C enclosing an open
surface can be thought of as the drawstring of a bag that can be
closed to create a closed surface.
Figure 1.5.2. Current density leaves a volume V and hence the net charge
must decrease.
Charge conservation, as expressed by (2), was a compelling reason for
Maxwell to add the electric displacement term to Ampère's
law. Without the displacement current density, Ampère's law
would be inconsistent with charge conservation. That is, if the
second term in (1) would be absent, then so would the second term in
(2). If the displacement current term is dropped in Ampère's law,
then net current cannot enter, or leave, a volume.
The conservation of charge is consistent with the intuitive
picture of the relationship between charge and current developed in
Example 1.2.1.
Example 1.5.1. Continuity of Convection Current
The steady state current of electrons accelerated through vacuum
by a uniform electric field is described in Example 1.2.1 by assuming
that in any plane x = constant the current density is the same. That
this must be true is now seen formally by applying the charge
conservation integral theorem to the volume shown in Fig. 1.5.3.
Figure 1.5.3. In steady state, charge conservation
requires that the current density entering through the x = 0 plane
be the same as that leaving through the plane at x = x.
Here the lower surface is in the injection plane x = 0, where the
current density is known to be Jo. The upper surface is at the
arbitrary level denoted by x. Because the steady state prevails, the
time derivative in (2) is zero. The remaining surface integral has
contributions only from the top and bottom surfaces. Evaluation of
these, with the recognition that the area element on the top surface
is (ix dydz) while it is (-ix dydz) on the bottom
surface, makes it clear that
This same relation was used in Example 1.2.1, (1.2.4), as the basis
for converting from a particle point of view to the one used here,
where (x, y, z) are independent of t.
Example 1.5.2. Current Density and Time-Varying Charge
With the charge density a given function of time with an axially
symmetric spatial distribution, (2) can be used to deduce the
current density. In this example, the charge density is
and can be pictured as shown in Fig. 1.5.4. The function of
time o is given, as is the dimension a.
Figure 1.5.4. With the given axially
symmetric charge distribution positive and decreasing with time
(p/t < 0), the radial current density is positive, as shown.
As the first step in finding J, we evaluate the volume
integral in (2) for a circular cylinder of radius r having z as its
axis and length l in the z direction.
The axial symmetry demands that J is in the radial direction and
independent of and z. Thus, the evaluation of the surface
integral in (2) amounts to a multiplication of Jr by the area
2 rl, and that equation becomes
Finally, this expression can be solved for Jr.
Under the assumption that the charge density is positive and
decreasing, so that do /dt < 0, the radial distribution of
Jr is shown at an instant in time in Fig. 1.5.4. In this case, the
radial current density is positive at any radius r because the net
charge within that radius, given by (5), is decreasing with time.
The integral form of charge
conservation provides the link between the current carried by a wire
and the charge. Thus, if we can measure a current, this law
provides the basis for measuring the net charge. The following
demonstration illustrates its use.
In Demonstration 1.3.1, the net charge is deduced from mechanical
measurements and Coulomb's force law. Here that same charge is
deduced electrically. The "ball" carrying the charge is stuck to the
end of a thin plastic rod, as in Fig. 1.5.5. The objective is to
measure this charge, q, without removing it from the ball.
Figure 1.5.5. When a charge q is introduced into an
essentially grounded metal sphere, a charge -q is induced on its
inner surface. The integral form of charge conservation, applied to
the surface S, shows that i = dq/dt. The net excursion of the
integrated signal is then a direct measurement of q.
We know from the discussion of Gauss' law in Sec. 1.3 that this
charge is the source of an electric field. In general, this field
terminates on charges of opposite sign.
Thus, the net charge that terminates the field originating from q
is equal in magnitude and opposite in sign to q. Measurement
of this "image" charge is tantamount to measuring q.
How can we design a metal electrode so that we are guaranteed
that all of the lines of E originating from q will be terminated
on its surface? It would seem that the electrode should essentially
surround q. Thus, in the experiment shown in Fig. 1.5.5, the charge
is transported to the interior of a metal sphere through a hole in its
top. This sphere is grounded through a resistance R and also
surrounded by a grounded shield. This resistance is made low enough
so that there is essentially no electric field in the region between the
spherical electrode, and the surrounding shield. As a result, there is
negligible charge on the outside of the electrode and the net charge
on the spherical electrode is just that inside, namely -q.
Now consider the application of (2) to the surface S shown in
Fig. 1.5.5. The surface completely encloses the spherical electrode
while excluding the charge q at its center. On the outside, it cuts
through the wire connecting the electrode to the resistance R. Thus,
the volume integral in (2) gives the net charge -q, while
contributions to the surface integral only come from where S cuts
through the wire. By definition, the integral of J
da over the cross-section of the wire gives the current i
(amps). Thus, (2) becomes simply
This current is the result of having pushed the charge through the
hole to a position where all the field lines terminated on the
spherical electrode.
3 Note that if we were to introduce
the charged ball without having the spherical electrode essentially
grounded through the resistance R, charge conservation (again applied
to the surface S) would require that the electrode retain charge
neutrality. This would mean that there would be a charge q on the
outside of the electrode and hence a field between the electrode and
the surrounding shield. With the charge at the center and the shield
concentric with the electrode, this outside field would be the same
as in the absence of the electrode, namely the field of a
point charge, (1.3.12).
Although small, the current through the resistor results in a
voltage.
The integrating circuit is introduced into the experiment in
Fig. 1.5.5 so that the oscilloscope directly displays the charge. With
this circuit goes a gain A such that
Then, the voltage vo to which the trace on the scope rises as the
charge is inserted through the hole reflects the charge q. This
measurement of q corroborates that of Demonstration 1.3.1.
In retrospect, because S and V are arbitrary in the integral
laws, the experiment need not be carried out using an electrode and
shield that are spherical. These could just as well have the shape
of boxes.
Charge Conservation Continuity Condition
The continuity
condition associated with charge conservation can be derived by
applying the integral law to the same pillbox-shaped volume used to
derive Gauss' continuity condition, (1.3.17). It can also be found
by simply recognizing the similarity between the integral laws of
Gauss and charge conservation. To make this similarity clear, rewrite
(2) putting the time derivative under the integral. In doing so,
d/dt must again be replaced by / t, because the time
derivative now operates on , a function of t and r.
Comparison of (11) with Gauss' integral law, (1.3.1), shows the
similarity. The role of o E in Gauss' law is played by J,
while that of is taken by - / t. Hence, by
analogy with the continuity condition for Gauss' law, (1.3.17), the
continuity condition for charge conservation is
Implicit in this condition is the assumption that J is
finite. Thus, the condition does not include the possibility of a
surface current.