Week 3: Potential Energy and Potential 109
that is large enough to contain sufficient charge for a smooth average charge density to result that
is also small enough that we can sum over it as if it is the integration volume elementdV (or ditto
for surface or linear distributions with elementsdAanddxrespectively).
Then the sum becomes:Vtot(~x) =∫
k dq 0
|~x−~x 0 |=∫
k ρ(~x 0 )d^3 r 0
|~x−~x 0 | volume (152)=∫
k σ(~x 0 )d^2 r 0
|~x−~x 0 |area (153)=
∫ k λ(~x
0 )dr 0
|~x−~x 0 |line (154)Deriving or Computing the Potential
The rules above give us two distinct ways to evaluate the potential inany given problem, and we
must look at the problem carefully to assess which one is best.
a) If the field is known, varies only in one dimension, and is integrable in some system of coordi-
nates, we can integrate
−∫
Exdxto find the potential. For all practical purposes in this course, problems involving the symmetric
distributions of charge whose fields we can find using Gauss’s Law areprecisely the ones where
it is likely to be most convenient to evaluate the potential in this way.
It isnecessaryto use this approach to find the potential differences of a non-compact charge
density distribution such as an infinite line or infinite sheet. This is because the sum of the
potential of an infinite amount of charge (however it is distributed)is infinite, which is in turn
why we restrict the use of the superposition forms of the potential that vanish at∞to compact
charge distributions.b) If the field is not known or discoverable from Gauss’s Law and/or isnot “one dimensional”
in the sense that we can easily find a line to integrate over where the vector components of
the field don’t enter in a non-trivial way, we will probably be better offcomputing the field
directly from the superposition principle – summing or integrating all of the contributions to
the potential from all the point charges or point-like elements of a charge distribution to find
the total.Note that both of these approacheswill yield the same answerfor charge distributions with
compact support within the inevitable constantV 0 forallproblems to which they are consistently
applied. In fact, even for non-compact distributions they will yield the same answer for the part
that varies with the coordinates of the point once one “renormalizes” the limiting form of the
superposition answer by subtracting the appropriate infinite constant. That’s because the negative
gradient of the two forms must, of course, return the same field!