W9_parallel_resonance.eps

Week 3: Potential Energy and Potential 123

z

σ

Figure 36: An “infinite” plane of uniform charge densityσ.

Potential of an Infinite Plane of Charge

Find the fieldandthe potentialrelative to the plane itselfat all points in space around an

infinite plane of charge. Explore the necessity of a finite referencepoint (where e.g.z= 0

is the most convenient) because the potential integratedinfrom∞is clearly infinite.

Using Gauss’s Law (or taking the limit of e.g. a disk on its axis) you can easily show that the
electric field a distancezabove an infinite plane of charge with charge densityσis:

Ez= 2πkeσ

(pointing away from the plane symmetrically on both sides) independent ofz. That is, the plane
of charge creates auniformelectric field that reaches from the plane to (in principle)∞without
change.

If we try to evaluate the potential at a finite pointzrelative to∞we get into trouble once again
because the charge distribution is non-compact:

V(z) =−

∫z

∞

2 πkeσ dz=∞− 2 πkeσz (186)

We feel uncomfortable with infinite quantities, so we either subtract away the infinity with a new
(infinite) constant of integration, or just measure the potentialdifference relative to some other zero.
A common, and convenient one (that leads to the same result as throwing away the infinity isz= 0,
on the plane itself. Interestingly, this is still well defined!

V(z) =−

∫z

0

2 πkeσ dz= 0− 2 πkeσz=− 2 πkeσz (187)

Again we will most often be interested in computing potential differences rather than potentials
in the subsequent chapters, especially for non-compact charge distributions. We note that the
functional variation withzis such that the potentialdecreaseswhen one moves away from the
plane; this is the most important thing to keep in mind when trying to assign or check the sign
of the potential (or potential difference). The fieldalwayspoints in the direction of decreasing
potential.

3.5: Conductors in Electrostatic Equilibrium

Last week we learned together, Gauss’s Law and the notion of equilibrium combine to give us impor-
tant information aboutconductors– material with an “inexhaustible” supply of charged particles
such as electrons that are free to move within the conductor and behave like an “electrical fluid”. In
particular, we determined thatE~= 0 inside a conductor in electrostatic equilibrium and thatE~||= 0
at the surface, so that any electrical field immediately outside its surface must be perpendicular to
the surface.

This suffices to show that conductors areequipotential– the potential difference between any two
points in the conductor or on its surface is:

∆V=−

∫~x 1

~x 0

E~·d~x= 0 (188)