Simple RC Circuit with a Resistor

The images below are visualizations of the electric charges and fields in a simple resistor-capacitor circuit. This circuit has additional resistance because the central region has a resistivity ten times higher than the other wires. 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.

(10 steps)

Note how the electric field in the middle of the bottom wire is starting to increase, and how two lines (sheets, actually) of charges are starting to form in the middle of the wire. These mark the boundaries the resistive wire segment.

The first simulation showed that the net charge in the interior of the wire is small (ideally zero) in steady-state, a consequence of Gauss's law and uniform electric fields. When two adjoining regions have different resistivities, however, charges will pile up at the interface. These charges alter the electric fields, increasing the electric field in the resistive region (thus increasing current flow) and decreasing the field and current in the low-resistance region. The later steps show these effects increasing until steady-state is reached.

(20 steps)
(40 steps)
(80 steps)
(160 steps)
(189 steps)

The simulation has nearly relaxed to steady-state. Note how the electric field in the central region is about ten times the field in the other regions. This stronger field compensates for the ten-fold increase in resistivity in this region, and results in equal amounts of charge moving into and out of the resistor.

This image shows the charges (red and blue, as before), the electric field in the space around the circuit, and equipotential surfaces (surfaces where the voltage is constant).

Norris Preyer