A Final Example

This last case differs because the surface charges have more work to do (or their effects are more dramatic), than the previous examples. The first circuit (Fig. 1) had the current basically parallel to the dipolar field of the capacitor, and the surface charges only made small modifications to the field. In Fig. 6, the dipolar electric field in the high-lighted region points opposite to the ultimate direction of current flow.

Figure 6: A sinuous resistor-capacitor circuit. All the wires are high-conductivity copper. The boxed region is discussed in the text.

Figure 7: The relaxation solution for the sinuous resistor circuit, from the initial conditions in (a) to nearly steady-state in (d). The panels are after 0, 10, 40, and 160 steps. The surface labeled $A$ is discussed in the text.

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Figure 7 shows the progress of the relaxation solution. We see polarization effects in panel (b), and see charges piling up along surface $A$, coming from both horizontal pieces of wire. In panel (c), we see the electric field in the highlighted region is beginning to change from the accumulating surface charges (along surface $A$ and elsewhere), and in panel (d), we see the electric field in the high-lighted region has reversed. This system is not yet fully relaxed, for the electric field does not yet have the same magnitude everywhere, but eventually it does (Fig. 6 is after another 200 relaxation steps).