Simple RC Circuit with Narrow Wire

The images below are visualizations of the electric charges and fields in a simple resistor-capacitor circuit. In this circuit, all the wires have the same resistivity, but the central section is narrowed, and so more resistive. The different images represent computational steps in the relaxation solution of the circuit.

The wires and plates 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 an 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 narrow wire is starting to increase, and how two lines (sheets, actually) of charges are starting to form on the ends of the wide wire. This is qualitatively similar to the resistive wire simulation, except here the cross-sectional area of the wire, rather than its resistivity, has changed.
	(20 steps)
	(40 steps)
	(80 steps)
	(160 steps)
	(200 steps) The simulation has nearly relaxed to steady-state. The large electric field in the narrow wire compensates for the smaller cross-sectional area and results in equal amounts of charge moving through all parts of the wire.