W9_parallel_resonance.eps

108 Week 3: Potential Energy and Potential

wherer=|~x|. Alternatively we could use the definition of the field relative to the force to define:

V(~x) =−

∫

E~·d~x+V 0 (146)

For charge distributions with compact support, we by convention pick the zero of potential at∞so
that:

V(~x) =−

∫~x

∞

E~·d~x (147)

In many cases (especially when we start to treat conductors morethoroughly in later chapters)
we will be interested inpotential differences. If the field is known and well behaved, they can be
easily computed by means of:

∆V(~x 1 →~x 2 ) =−

∫~x 2

~x 1

E~·d~x (148)

We can invert these relations to obtain:

E~=−∇~V (149)

which in some cases will give us a relatively easy path to find the field. Ifthe potential is relatively
easy to find by (say) superposition (because it is a straight scalar sum or integral over the potentials
of all the contributing charges) then one can find the field by doing relatively easy derivatives instead
of sums or integrals over vector components.

Note that this relation gives us a new way to write the strength of a field in SI units as volts per
meter. Note also that there is a precise analogy between force andpotential energy and field and
potential. Finally, note that once we know the potential produced by a collection offixedcharges, we
can compute the potential energy of a chargeqplaced in the potentialsubject to the conditionthat
the presence of the charge in the potential does not cause significant rearrangement of the charges
that create that potential as:
U=qV (150)

This will not always be the case! In fact, if we were picky we’d say thatit is almost never the
case in nature, because atoms aren’t “solid” objects and inevitablydistort in the presence of the
field of the perturbing charge. However, that doesn’t really stop us from using this expression; we
merely have to compute the potential energy in theself-consistentperturbed potential of the other
charges. It does make it a bit more difficult, though.

3.3: Superposition

As we noted in the previous section, a major motivation for introducing potential is that it is a scalar
quantity that we can evaluate by doing sums that don’t involve the complexity of vector components
or charge-charge interactions. The rule for finding the potentialof a collection of charges is simple:
We just add up the scalar potential of each (point-like) charge independent of all the rest!

This is once again thesuperposition principlefor electrostatics, now applied to the scalar poten-
tial:
Vtot(~x) =

∑

i

kqi

|~x−~xi|

(151)

In words, the potential at a point in space is the simple (scalar) sum of the individual potentials of
all the charges that contribute to that total potential.

As before, when we are working at scales where there are many many elementary point charges
contributing to the potential, we can coarse grain average. That is, we can look at a volume ∆V