The Height Metaphor

You might be familiar with topographic maps: they are maps which show how the elevation changes in a particular area. They do so by means of contour lines, which connect points of equal elevation, and are usually labelled with the distance they are above sea level in feet. The topographic map here shows the peaks of two hills, indicated by the closed contour lines which have the highest elevation of all of them shown. It also shows a valley on the lefthand side, or a "trough", where the elevation is a minimum. If a boulder were placed on this map, assuming there were no trees or small bumps etc, it would roll away from the hills, downhill, and into the trough (or some other low point). Equipotential lines are a lot like contour lines: they connect points of equal potential, and indicate the "shape" of the field that surrounds charges. They also indicate how a target charge will react when placed there, just as contour lines indicate how a boulder will roll.

In fact, it is quite useful to talk about potential as if it were like height. It's not a perfect metaphor, because potential exists in three dimensions while elevation only exists in two, but nevertheless it can help give us an intuitive understanding of how potential works.

Source Charges

This figure shows the graph of the potential for a point charge as a function of distance. The upper blue curve is the graph for a positive charge: it gets higher and higher as one moves closer and closer to the charge, as if the positive source charge were a mountain. A negative source charge however, represented by the lower curve, is like a pit or trough: the potential gets lower and lower as one approaches it. Both potentials approach zero as one moves far away from the source charge, which might be analogous to "sea level".

Target Charges

Positive target charges always feel a force towards lower potential: in other words, they "roll downhill" like a boulder would. That means that a positive target near a positive source will roll away from the peak, while a positive target near a negative source will roll into the pit. Negative targets do the opposite: they "roll uphill".

Direction and Strength of the Force

If you place a boulder on the side of a hill, assuming no obstacles in the way, it will roll directly downhill, which is always a direction perpendicular to the contour lines. Also, the closer the contour lines are together, the steeper the terrain is (you're changing elevation over a shorter horizontal distance), which means a boulder will feel a greater force and roll faster. The same thing holds true for equipotentials. The force a charge feels is always perpendicular to the equipotential it is on, regardless of the locations of the source charges: if you have the equipotentials you don't need Coulomb's Law to figure out the direction. Similarly, places where the equipotential lines are closer together are places where charges will feel a stronger force. We will quantify this a bit more when we talk about electric field in a later chapter.

The Baseline

While it is customary to measure all elevations from sea-level, this is just a custom. Alternative choices could be imagined. For example, a hiker in Colorado might want to measure all heights relative to the altitude of Denver (5280 feet) as I've done on the map shown here: this could be in order to make the numbers smaller, to make it easier to know where they are relative to the city, or for other reasons. The alternative set of numbers don't really change anything about the map (why should someone in Denver care about sea level, after all?) because the differences in elevation are what really matter, not the actual values. Similarly, it is only differences in potential that have real physical meaning, because they are connected to changes in potential energy, and only changes in energy have real meaning. Thus while it is customary to say that the potential very far away from all charges is 0V, it is not necessary, and in fact we can place our baseline to be anywhere we want, by saying that any one equipotential in our diagram is "0V". Now mind you, we should not do this if we are calculating the potentials of point charges or using the formula $V=k{q_s\over d}$, because it introduces complications we don't want to deal with. But when we think purely about equipotentials and target charges, or (as we will see) when we think about conductors and capacitors and currents, then we are truly at liberty to place our baseline wherever we wish.