Two or Three Dimensions

(a) A 19th century USGS topographical map of Shelburne Falls, Mass.

The topographical map shown in figure (a) suggests a good way to visualize the relationship between field and voltage in two dimensions. Each contour on the map is a line of constant height; some of these are labeled with their elevations in units of feet. Height is related to gravitational potential energy, so in a gravitational analogy, we can think of height as representing voltage. Where the contour lines are far apart, as in the town, the slope is gentle. Lines close together indicate a steep slope.

If we walk along a straight line, say straight east from the town, then height (voltage) is a function of the east-west coordinate x. Using the usual mathematical definition of the slope, and writing V for the height in order to remind us of the electrical analogy, the slope along such a line is ΔV/Δx, or dV/dx if the slope isn't constant.

What if everything isn't confined to a straight line? Water flows down hill. Notice how the streams on the map cut perpendicularly through the lines of constant height.

(b) The constant-voltage curves surrounding a point charge. Near the E = -dV/dz charge, the curves are so closely z spaced that they blend together on this drawing due to the finite width with which they were drawn. Some electric fields are shown as arrows.

1V 2V 3V It is possible to map voltages in the same way, as shown in figure (b). The electric field is strongest where the constant-voltage curves are closest together, and the electric field vectors always point perpendicular to the constant-voltage curves. The figures on the following page show some examples of ways to visualize field and voltage patterns. Mathematically, the calculus of the preceding section generalizes to three dimensions as follows:

E _x = -dV/dx

E _y = -dV/dy

E _z = -dV/dz

Self-Check

Imagine that the topographical map represents voltage rather than height. (a) Consider the stream the starts near the center of the map. Determine the positive and negative signs of dV/dx and dV/dy, and relate these to the direction of the force that is pushing the current forward against the resistance of friction. (b) If you wanted to find a lot of electric charge on this map, where would you look?

Answer

(a) The voltage (height) increases as you move to the east or north. If we let the positive x direction be east, and choose positive y to be north, then dV/dx and dV/dy are both positive. This means that Ex and Ey are both negative, which makes sense, since the water is flowing in the negative x and y directions (south and west). (b) The electric fields are all pointing away from the higher ground. If this was an electrical map, there would have to be a large concentration of charge all along the top of the ridge, and especially at the mountain peak near the south end.

Two-dimensional field and voltage patterns.

Top: A uniformly charged rod. Bottom: A dipole.

In each case, the diagram on the left shows the field vectors and constant-voltage curves, while the one on the right shows the voltage (up-down coordinate) as a function of x and y.

Interpreting the field diagrams: Each arrow represents the field at the point where its tail has been positioned. For clarity, some of the arrows in regions of very strong field strength are not shown - they would be too long to show.

Interpreting the constant-voltage curves: In regions of very strong fields, the curves are not shown because they would merge together to make solid black regions.

Interpreting the perspective plots: Keep in mind that even though we're visualizing things in three dimensions, these are really two-dimensional voltage patterns being represented. The third (up-down) dimension represents voltage, not position.