Introductory Physics Volume Two

1.5 Continuous Charge Distributions 17
§ 1.5 Continuous Charge Distributions
In principle one could compute the electric field in any situation
by summing the fields of each proton and electron in the system. In
practice this is not done because there are too many electrons and
protons. In this section we will see how to compute the electric field
when there are too many charges to count.
Suppose that we have a string that has been rubbed on a cat until
the string has built up a charge Q that is spread uniformly over the
length of the string. We then take the string and stretch it out in a
straight line. We wish to calculate the electric field due to the string.
For convenience assume that the string is stretched out along the x-axis
from x = −L/2 to x = L/2.
One way of finding the electric field is to conceptually break the
string into short little sections, say N of them. Each of the sections
will be a length dx = L/N, and carry a charge dq = Q/N. We can
number the sections from 1 to N, and then the n’th section will be
at the position ⃗r n = x n î (x n = nL/N − L/2). As long as we break
the string into enough sections so that dx is small, we can treat each
section as a point charge and then we can compute the electric field as
a sum of N point charges
⃗E(⃗r) = ∑ q n ⃗r − ⃗r n
4πɛ
n 0 |⃗r − ⃗r n | 3 = ∑ dq ⃗r − ⃗r n
4πɛ
n 0 |⃗r − ⃗r n | 3
Keep in mind when you look at this formula that the vector ⃗r n points
to the charge q n . With this formula and the aid of a computer it
is possible to compute the electric field for nearly any distribution of
charge that you can imagine.
It is also possible, in some cases, to find a closed form solution,
without using a computer. If we take the limit as N goes to infinity,
the sum becomes an integral and the electric field can be written in the
form
∫
dq ⃗r − ⃗r s
⃗E(⃗r) =
4πɛ 0 |⃗r − ⃗r s | 3
Here again we imagine that we break the object into many small pieces
of charge dq, the vector ⃗r s points toward dq, ranging over all dq’s and