Lab 6
Equipotentials

Equipotentials are regions of constant electric potential. They can be found by experiment or by computer simulation. The experimental procedure is quite simple. A scale model of an electrode assembly is constructed in an insulating container called an electrolytic trough. The container is filled with water and regions of constant potential are measured with a voltmeter on a grid of closely spaced points. The deflection of an electron beam in the first television picture tube was modeled in an electrolytic trough.

The goal today is to use this procedure to model the deflection of an electron beam between two parallel, conducting plates in vacuum. The electrons emerge from a source of electrons called an electron gun, represented by a third plate. A printed circuit board applies a voltage to each plate. See Figure 1.

Figure 1. An Electrolytic Trough

Review Theory

Regions of constant electric potential in two dimensions define a line called an equipotential. Equipotentials are important because they can be used to predict the trajectory of charged particles. A charged particle will always experience a force in a direction perpendicular to an equipotential.

Imagine a two dimensional grid of equally spaced points of known potential. If the points are closely spaced, the potential between the points will vary linearly with the distance between them. Consider two adjacent points (x0, y1) and (x0, y2) where the potential is V1 and V2, respectively. The potential, V, at any point between them (x0, y0) can be found by interpolation, since:

From which:

An equipotential can be found by repeating the interpolation process for the potential, V, at each value of x in the grid. In three dimensions equipotentials form an equipotential surface.

In practice, voltages (or potential differences) are actually measured. The potential, V, is found if the voltage is always measured with respect to a common or reference potential. The reference potential is arbitrary, but it is often chosen to be ground (the potential of the earth where the measurement is made).