a/A topographical map of
Shelburne Falls, Mass.(USGS)b/The constant-potential curves
surrounding a point charge. Near
the charge, the curves are so
closely 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.everyday life we should expect it to be very difficult to detect any
effect from the weight or inertia of an electron.
As an example, suppose that a metal rod of lengthLis oriented
upright. The conduction electrons are free to move, so they would
tend to drop to the bottom of the rod. Electrical forces will however
resist this segregation of positive and negative charges. To esti-
mate how hard it would be to observe such an effect, let us imag-
ine connecting the probes of a voltmeter to the ends of the rod.
In equilibrium, the electrical and gravitational fields must have ef-
fects on an electron that cancel out. Setting the magnitudes of
these forces equal to each other, we haveeE=mg, and since
E=∆V/L, we predict a voltage difference∆V= (m/q)gL. For a
one-meter rod, the predicted effect is∼ 10 −^10 V.
This is quite small, but not impossible to measure, and the the-
oretical prediction was confirmed for a similar experiment by Tol-
man and Stewart in a 1916 experiment at Berkeley. This was the
first direct evidence that the charge carriers inside a metal wire
are in fact electrons. Similarly, we do expect mechanical side-
effects in any electrical circuit, e.g., a slight twitching of a flash-
light when we turn it on or off, but these will be much too small
to notice except with exceptionally delicate and sensitive tools. It
is surprising that we can get information about the microscopic
structure of a metal merely by measuring its properties in this
way. Another, similar example along these lines is described in
sec. 11.2.4, p. 695.10.2.2 Two or three dimensions
The topographical map in figure a suggests a good way to visu-
alize the relationship between field and potential 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 energy, so in a gravitational analogy, we can think
of height as representing potential. 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 (potential) is a function of the east-west coordinatex.
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 dV/dx(the rise over the run).
What if everything isn’t confined to a straight line? Water flows
downhill. Notice how the streams on the map cut perpendicularly
through the lines of constant height.
It is possible to map potentials in the same way, as shown in
figure b. The electric field is strongest where the constant-potential
curves are closest together, and the electric field vectors always point
Section 10.2 Potential related to field 593