Week 3: Potential Energy and Potential 125
Now, however, we compute theradial field at the surfaceof the two conductors. It is:Ea =kq
a^2 (194)
Eb =kq′
b^2 (195)If we take the ratio of thefield strengthswe get:
Ea
Eb=q
q′b^2
a^2=b
a(196)
and conclude that thefield is much stronger on the surface of the smaller conductor. In fact, it
becomesinfinitein the limit thata→0 relative to a finiteb.
What this tells us is that the field in the vicinity of a conductor in electrostatic equilibrium at
some non-zero potential ismuch stronger at sharp pointsthan it is on smooth surfaces with a large
radius of curvature. This has important consequences, as we shall see!
3.6: Dielectric Breakdown
Insulators are not ever perfect, because electrons as charge carriers are not bound to the conducting
substrate by an infinite potential energy barrier. In a sufficiently large field electrons are torn
from their parent atoms and insulators “suddenly” become conductors, a process calleddielectric
breakdown. Lightning is a spectacular example of dielectric breakdown in nature.
The way lightning (or any sort of arc discharge) works is that charge builds up on clouds and/or
the ground to create a large potential difference. At some point the field strength associated with
this potential difference becomes great enough that the force it exerts on electrons exceeds the
force binding the electrons to their parent atoms in the insulator (or alternatively, they get enough
potential energy to overcome the potential energy barrier thatconfines them). At first only a few
electrons get away, and are quickly accelerated by the field as theyget over the confining potential
barrier.
These electrons in turn collide with other nearby atoms, tranferring momentum to them and
knocking still more electrons loose. A cascading chain reaction occurs that heats the atoms in the
path of the ever increasing flow of charge and knocks still more charge loose to join that flow. In a
fraction of a second, the superheated air becomes a white-hotplasmathat conducts electricity quite
well and the enormous charge difference between ground and cloudor cloud and cloud neutralizes
in a burst of millions of ampere’s of current. Bang! Zap! Ouch!
It is important to remember whenever working with high voltages thatfew materialsare terribly
good insulators against the strong fields associated with large potential differences over a short
distance. That is, if you get close enough to a high voltage line it will simply arc over and electrocute
you. It may well arc through a piece of glass or plastic and kill you. Wood is an insulator for ordinary
voltages but conducts more than enough to kill you if you try to touch a high voltage power line
with a stick.
Note also that if one approaches a conductor with a charge, oneinducesa charge on the part of
the conductor nearest the charge. If that part happens to be asharp point, the properties of charge
sharing on an equipotential conductor create anextremely strong fieldin the immediate vicinity of
the point. The field at a sharp point can easily be strong enough to ionize air molecules in the
immediate vicinity of the tip and make them conduct! The ionized air molecules recover electrons
from their surroundings, which emit light as they rebind. This light (visible in the dark as a faint
blue-violet glow on a thumbtack point attached to an electrostatic generator) is called thecorona.