42 Week 1: Discrete Charge and the Electrostatic Field
superposition principle inintegralform. Note that the result of this sort of computation willfailif
we examine~rinside the material itself very close to one of the consituent discrete charges (where
the 1/r^2 nature of the force guarantees that if you are close enough to a charge, its field will
overwhelm the field of all more distant charges) but in general the resulting numbers are both useful
an remarkably accurate, accurate as an “average value” even within a material.
dE
r 0
r
r r 0
dq=ρdV 0
y
x
Figure 3: The geometry needed to evaluate the field of a general continuous charge distribution.
Note well the similarity to the geometry for a collection of charges, except that the many “point
charges” are all chunks of differential volume with chargedqand the “sum” is now an integral.
To write down the integral (and help us remember it) we begin by usingthe basic equation
obtained above for the field of a point charge and apply it to a tiny “chunk” of the charge distri-
butiondq– one small enough to be considered a point-like charge. We write thisas thedifferential
contribution of the charge to the overall field as follows:
dE~(~r) =kedq(~r−~r^0 )
|~r−~r 0 |^3(8)
We then use one of the definitions of charge density to convertdqinto e.g. dq=ρ dV 0 =
ρ(~r 0 )d^3 r 0 :
dE~(~r) =keρ(~r 0 ) (~r−~r 0 )d^3 r 0
|~r−~r 0 |^3(9)
Finally, we integrate both sides of this equation over the entire volumeV whereρ(~r 0 ) is sup-
ported. The resulting integral form is:
E~(~r) =ke∫
Vρ(~r 0 )(~r−~r 0 )dV 0
|~r−~r 0 |^3(10)
for a 3-dimensional (volume) charge distribution,
E~(~r) =ke∫
Sσ(~r 0 )(~r−~r 0 )dS 0
|~r−~r 0 |^3(11)
for a surface charge distribution on a surfaceS, and
E~(~r) =ke∫
Lλ(~r 0 )(~r−~r 0 )dL 0
|~r−~r 0 |^3