W9_parallel_resonance.eps

Week 1: Discrete Charge and the Electrostatic Field 41

1.4: Superposition Principle

E 3

y

x

r

q

r

q 2

q

3

r

r 3

P

r r 3

2

1

1

Figure 2: Geometry needed to evaluate the field of many charges. Only the field of the third charge
E~ 3 is shown explicitly. Note well the magnitude and direction of the vector~r−~r 3 – head at~r, tail
at~r 3. This is a vectorfromthe position of the chargeq 3 tothe point of observationP at~r.

Given a collection of charges located at various points in space, the total electric field at a point
is the sum of the electric fields of the individual charges:

E~(~r) =

∑

i

kqi(~r−~ri)

|~r−~ri|^3 (7)

Simple as it is, the superposition principle isextremely importantin physics. It tells us that the
electrostatic field results from alinearfield theory and later in a study of physics you will learn that
this means that the differential equations that describe the field arelineardifferential equations.

Note that it doesn’t have to be that way. There is nothing inherentlycontradictory about two
charges producing a field at a point in space that islessthan their sum ormorethan their sum. There
are examples in physics of interactions that do just that (althoughthis sort of complication, like the
“three body forces” that are also excluded by linearity, makes thetheoriesmuchmore difficult to
solve).

In pure classical physics the field is strictly linear, but in quantum theory the electromagnetic
field becomes (in a sense)nonlinearat very short distances from elementary charges due to vacuum
polarization and in just the right way to “soften” the singularity in certain interactions and be
unified with other forces of nature in a single field “theory of everything”. Inthiscourse, however,
we will never ever explore the quantum distance or interaction scales where this sort of thing is an
issue, so for us superposition will be a fundamental principle.

As noted above, charge, while discrete, comes in very tiny packages of magnitudeesuch that
matter contains order of 10^27 charges per kilogram, with roughly equal amounts of positive and
negative charge so that most matter is approximately electrically neutral most of the time. When
we consider macroscopic objects – ones composed of these enormous numbers of atoms and charges

• it therefore makes sense to treat the distribution and motion of charge as if it iscontinuously
  distributed.

In order to find the electrostatic field produced by a charge density distribution, we use the