W9_parallel_resonance.eps

124 Week 3: Potential Energy and Potential

Note that this doesn’t mean that the potential of the conductor iszero, only that it is aconstant.
That is consistent:
E~=−∇~V 0 = 0 (189)

whenV 0 is any constant.

This also permits us to make an important observation. For any arrangement of (say two)
isolated conductors with sufficient symmetry that we can put an arbitrary charge on either of them
and not have their interaction break the symmetry of the charge’sredistribution, we can compute
thepotential differencebetween the conducting pair as a function of the charge differencebetween
them. This potential difference will turn out to be proportional to the charge transferred and will
only otherwise depend on thegeometryof their arrangment. In the next chapter this will be the
basis of the notion ofcapacitance.

Charge Sharing

b

a

+Q

Figure 37: Charge sharing between two distant conductors connected by a wire. They become
equipotential, with charge transferred (shared) between them to make it so.

Here is an important example of equipotentiality. Suppose one has two conducting spheres, one with
radiusaand one with radiusbsuch thata≪b(as seen in figure??above. Let us further suppose
that the spheres are very distant from one another so that the field of one is very weak in the vicinity
of the other (so that very little charge redistribution occurs if oneor the other is charged up). We
begin by imagining that we have put a chargeQon sphereb.

In that case it is easy to see or show that:

Vb=−

∫b

∞

Erdr=

kQ

b

(190)

everwhere inside spherebwhile
Va= 0 (191)

on the other sphere. There is clearly a potential difference between the two spheres. Now imagine
that we connect the two with a thin conducting wire. They form a single conductor and therefore
quicklyequalizetheir potentials as charge flows frombtoa.

Charge is conserved. They will reach equilibrium when:

k(Q−q)

b

=

kq′

b

=

kq

a

(192)

whereqis the net charge transferred frombtoaandq′is the remaining charge onb. This can be
rewritten as:
q
q′

=a

b

(193)

The smaller the sphere the smaller the fraction of charge on it, whichmakes sense since theratioof
charge to radius must be the same.