The Simplest Circuit

The first circuit (see Fig. 1, showing the fully relaxed solution) is the simplest resistor-capacitor ($RC$) circuit, with fat copper wires connecting the copper plates of the capacitor. The diagram is a slice through the mid-plane of the circuit, with the colors represent charges, red for positive and blue for negative. The scale was chosen to show the surface charges, and the charges on the plates are much higher (about $10^5$ elementary charges/cell).

Figure 1: This is a slice through the mid-plane of a simple resistor-capacitor circuit. The colors represent the amount of excess elementary charges in a computational cell (the charges on the inner surfaces of the capacitor plates are off-scale). The arrows represent the electric field at that point, scaled by taking the square-root of the magnitude. The large field between the plates has not been plotted.

The arrows represent the electric field, created by all the charges, and are scaled by plotting the square-root of the magnitude of the field. The very large uninteresting electric field between the plates is not shown. Note that outside the circuit, the field looks like the dipolar field of the capacitor, but inside the circuit the electric field is uniform in magnitude and parallel to the wires. It wasn't always like that--Fig. 2 shows the sequence of relaxation steps (the pictures are after 0, 10, 40, and 160 steps; see Preyer[7] for details).

Figure 2: These diagrams illustrate the relaxation of the solution, from the initial conditions in (a) to the steady-state solution in (d). The panels are after 0, 10, 40, and 160 steps. For clarity, the arrowheads on the electric field vectors are not plotted.

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Figure 2(a) shows the initial configuration, with excess charges only on the capacitor plates. The electric field everywhere is that of the finite capacitor. The second panel (Fig. 2(b)) shows two major effects: polarization of the wires from the dipolar field, and charges on the outer surfaces of the capacitors.

Recall that the electric field shows the direction of force on positive charges; if the electric field points towards some surface, that surface becomes positive. Other surfaces will become negative because the electric field points away from them. The interior remains neutral, for as many charges move into some small volume as move out (except where there are changes in the conductivity--see the next section).

The charges on the outside of the plates are the result of the finite size of the plates. The first panel shows the electric field outside the two sheets of charge is not zero (it is zero only for infinitely large parallel plates), but rather acts to push some charges away from inner surfaces and on to the outer surfaces or out into the circuit.

The remaining two panels shows the approach to the fully relaxed solution. Note surface charges and interior electric field are both changing, with the next effect of producing a nearly uniform electric field everywhere parallel to the wires.

Focus on the upper-left corner of the circuit. In panels (a) and (b) the electric field points into this corner. Positive charges (speaking in terms of conventional current) are pushed into the corner, making it more positive. As it becomes more positive, it begins repelling positive charges, and by panel (c) the electric field no longer points into the corner, but in the direction of current flow. Similar feedback processes are occurring throughout the circuit: if the electric field in some region is not the steady-state field (the one that produces uniform charge flow), charges will be moved to the surface in such a way as to correct the electric field. By the final panel (Fig. 2(d)), the electric field is fairly uniform throughout the circuit.