a/Example 12.c/Two-dimensional field and po-
tential patterns. Top: A uniformly
charged rod. Bottom: A dipole.
In each case, the diagram on the
left shows the field vectors and
constant-potential curves, while
the one on the right shows the
potential (up-down coordinate) as
a function of x and y. Interpret-
ing the field diagrams: Each ar-
row represents the field at the
point where its tail has been po-
sitioned. 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-potential
curves: In regions of very strong
fields, the curves are not shown
because they would merge to-
gether to make solid black re-
gions. Interpreting the perspec-
tive plots: Keep in mind that even
though we’re visualizing things
in three dimensions, these are
really two-dimensional potential
patterns being represented. The
third (up-down) dimension repre-
sents potential, not position.10.3 Fields by superposition
10.3.1 Electric field of a continuous charge distribution
Charge really comes in discrete chunks, but often it is mathe-
matically convenient to treat a set of charges as if they were like a
continuous fluid spread throughout a region of space. For example,
a charged metal ball will have charge spread nearly uniformly all
over its surface, and for most purposes it will make sense to ignore
the fact that this uniformity is broken at the atomic level. The
electric field made by such a continuous charge distribution is the
sum of the fields created by every part of it. If we let the “parts”
become infinitesimally small, we have a sum of an infinitely many
infinitesimal numbers: an integral. If it was a discrete sum, as in
example 3 on page 584, we would have a total electric field in thex
direction that was the sum of all thexcomponents of the individual
fields, and similarly we’d have sums for theyandzcomponents. In
the continuous case, we have three integrals. Let’s keep it simple by
starting with a one-dimensional example.
Field of a uniformly charged rod example 12
.A rod of lengthLhas chargeQspread uniformly along it. Find
the electric field at a point a distancedfrom the center of the rod,Section 10.3 Fields by superposition 595