Resistors

The next simplest circuits are ones with added resistors. Figure 3 has a region with ten times the resistivity (or one-tenth the conductivity) of the other copper wires, and Fig. 5 has a narrow copper wire.

Figure 3: This circuit has a resistive region in the bottom wire, where the conductivity is one-tenth that of the other wires.

Figure 4: The relaxation solution for the added-resistor circuit, from the initial conditions in (a) to the steady-state solution in (d). The panels are after 0, 10, 40, and 160 steps.

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The first resistor circuit shows a new phenomena: there are excess charges at the interface between the wire and the resistor. The origin of the charges is easily determined by examining the steps in the relaxation solution (Fig. 4). Looking at panels (a) and (b), we see the electric field of the capacitor has a component to the right in the middle of the lower wire. Focus on the left-hand boundary of the resistive region: the electric field is essentially the same on both sides of the boundary, but the effect is not the same. From Eq. 1, the electric field pushes ten times more positive charges on to the left-hand side of the boundary than away from the right-hand side of the boundary. Thus positive charges pile up at this boundary (and negative charges on the other boundary), and it looks like a second capacitor. These charges act to increase the electric field in the low-conductivity region, and decrease the field in the high-conductivity region, until steady-state has been reached (panel (d)).

Figure 5 is similar, except here the resistance is caused by a physical narrowing of the wire. Again, capacitor-like charges build up, increasing the field in the resistive region, and decreasing the field elsewhere in the circuit, until the same current flows throughout the circuit.

Figure 5: The added resistor is now a narrow region of wire.