3.3.2 A capacitor with time-varying charges on its plates
Figure 4 shows a parallel plate capacitor with circular plates, which is being charged by steady currents flowing along straight wires. We know that there is a circular pattern of magnetic field lines around the wires, but what happens inside the capacitor, between the plates?
The situation illustrated in Figure 4 is difficult to analyse quantitatively. Charge spreads out over the plates from the points of contact with the wires so, at any given moment, the plates are unevenly charged and radial currents flow over their surfaces. We will avoid such complications by imagining that the charge is conveyed by a uniform steady current density that is perpendicular to the full area of the plates. One way of approaching this ideal would be to replace the arrangement of Figure 4 by thick cylinders separated by a narrow gap, as in Figure 5. The gap between the cylinders is tiny compared to their diameters, so the system behaves like an infinite parallel plate capacitor, with the end-faces of the cylinders serving as the capacitor plates.
Between the plates, there is no charge flow so the current density J is equal to zero. However, the Maxwell term is non-zero there because the increasing charge on the plates produces a steadily increasing electric field between the plates. Taking the gap between the plates to be tiny (so that we can ignore edge effects), the electric field between the plates is uniform and has the instantaneous value
where Q(t) is the instantaneous charge on the positive plate, A is the area of a plate and ez is a unit vector pointing from the positive plate to the negative plate. The Maxwell term in the gap is
so the differential version of the Ampère–Maxwell law in the gap is
The corresponding integral equation is
where S is an open surface and C is its perimeter.
Exploiting the axial symmetry of the situation, we use cylindrical coordinates with the z-axis along the line of symmetry. We also assume that the magnetic field has the form
For the moment, we have allowed for a possible dependence of Bφ on z. This is a wise precaution because the present situation does not have translational symmetry, but you will soon see that it is not necessary.
To apply the Ampère–Maxwell law, we choose the circular path C shown in Figure 6, together with the disc S that has C as its boundary. Equation 7.14 then gives
and the magnetic field between capacitor plates is
This is independent of z, and is also independent of time because we are assuming that the capacitor is being charged at a constant rate by a steady current.
I should perhaps point out that I am not claiming that the Maxwell term causes the magnetic field inside the capacitor. It would be silly to neglect the currents that bring charge to the capacitor plates. These currents do not flow inside the capacitor, but there is nothing to prevent them from producing a magnetic field inside the capacitor. Indeed, if the gap between the plates is small, the magnetic field inside the capacitor due to external currents must overwhelm anything else. This may lead you to wonder why the above calculation, based on the Maxwell term, is valid. The logic is as follows. First, the time-varying charges on the capacitor plates produce a time-varying electric field between the plates. Then the Ampère–Maxwell law provides a relationship between the time-varying electric field and the circulation of the magnetic field. This relationship must be satisfied by all electric and magnetic fields, and it allows us to deduce the magnetic field from the known electric field irrespective of what the causes of these fields might be.
It is also instructive to calculate the magnetic field inside the capacitor by an alternative route. Instead of choosing S to be a disc, we can take it to be the open cylinder shown in Figure 7, with its end-cap outside the capacitor. The unit normal to the end-cap is chosen to point along the positive z-axis, in accordance with the usual right-hand grip rule.
Outside the infinite parallel plate capacitor, there is no time-dependent electric field, so there is no Maxwell term. However, there is the steady uniform current density that brings charge to the capacitor plates. This current density obviously obeys
Now, both the Maxwell term inside the capacitor and the current density outside the capacitor are perpendicular to the capacitor plates (remember, we have carefully avoided any radial flow of current). So, if we apply the integral version of the Ampère–Maxwell law (Equation 7.11) to the surface in Figure 7, the curved sides of the cylinder contribute nothing, and we are left with an integral over the end-cap. The Ampère–Maxwell law then gives
exactly as before. This shows why the Maxwell term is needed. Without it, these two methods of calculating the magnetic field inside the capacitor would give different answers, leading to a contradiction. Very similar calculations show that the magnetic field outside the capacitor is given by exactly the same expression, so there is no difference between the magnetic field inside and outside the capacitor.
Finally, it is interesting to note that the predictions of the Ampère–Maxwell law can be put to a direct experimental test. In 1973, Carver and Rajhel carried out a demonstration using the apparatus sketched in Figure 8. An oscillating voltage was applied across the circular plates of a large parallel plate capacitor, producing an oscillating electric field inside the capacitor. From the above argument, we would expect this to be accompanied by an oscillating Bφ field. The toroidal coil in Figure 8 was designed to detect this. The oscillating magnetic flux through the toroidal coil induced an oscillating voltage, which was easily detected on an oscilloscope.