Simple RC Circuit

The images below are visualizations of the electric charges and fields in a simple resistor-capacitor circuit. The resistor is a uniform copper wire joining the two plates. The different images represent computational steps in the relaxation solution of the circuit.

The wires and plates are divided into computational cells, each a cube 0.25 mm on a side. The entire circuit is 25 mm (about an inch) across, the wires are 5 mm thick, and this image is a slice through the mid-plane of the circuit.

The colors represent the amount of excess charge in each cell, from red (1000 or more positive elementary charges), to white (neutral), to blue (1000 or more negative elementary charges). The arrows show the magnitude and direction of the electric field (due to all the charges) calculated at that point.

Click on an image for a larger, clearer picture (800x600 pixels, about 25k each).

This shows the capacitor at t=0, when charges are placed on the inner surfaces of the left- and right-hand plates. The white color indicates no excess charges are present elsewhere in the circuit. The field vectors are very large near the plates, and elsewhere look like the field of a finite electric dipole.
(5 steps)

We see polarization effects on the inner surfaces of the wire, but the electric fields are hardly changed yet.

Note how there are surface charges on both sides of each capacitor plate; these important charges help create the fields that push charges away from the opposite-sign charges on the other plate. The simplest model of very large parallel-plate capacitors would predict zero electric field outside the capacitor plates, so we wouldn't expect any charges to move!

(10 steps)

Surface charges are starting to pile up in the corners--especially the upper left and upper right corners. The first charges moving away from (or towards) the capacitor encountered no obstacles and moved straight into (or from) the walls. The motion of later charges is affected by these surface charges, and we see this by the changing electric field vectors.

(20 steps)

Notice how the electric field vectors are becoming more uniform in magnitude and parallel to the wires. In steady-state, we expect the same current in all parts of the circuit, which requires the same electric field in all parts of the circuit. Other simulations explore circuits where the electric field is not the same in all places when in steady-state.

(40 steps)
(80 steps)
(160 steps)
(200 steps)

The electric field is now very uniform and parallel to the wires, everywhere in the circuit, and the circuit is nearly relaxed to steady-state. Subsequent evolution is boring: the capacitor slowly discharges, and all the surface charges slowly diminish to zero.

There is an important physical principle evident here: the charges are all on the surface of the wire due to Gauss's Law. Gauss's law relates the net number of electric field lines that cross a closed surface to the charge inside that surface (field lines that point out count as positive, lines that point in count as negative). Since the electric field is uniform in magnitude and direction, the net number of field lines passing through the surface of a little cube (or other shape) is zero, so there's no charge inside the little box. The only place this argument could fail is at the surface of the wire, and that's the only place we find excess charges!

Norris Preyer