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Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19
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Summary: The Classical Electron;
Conductors
1. The Classical Electron
2. Conductors
3. Capacitance
Suggested Reading:
Griffiths: Chapter 2, Section 2.5, pages 96-109.
Wangsness: Chapter 6, pages 83-97; Chapter
7, Section 7.4 pages 103-109.
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The Classical Electron
In classical electromagnetic theory one can model
the electron as a sphere with a finite radius r0,
a charge q = −1.6 × 10−19C and a mass me =
9.11 × 10−31kg (or roughly 0.5 MeV). Experimentally it is known that at least some of the mass is
electromagnetic in origin. We know this because
the electromagnetic field of a moving charge has
a momentum which is proportional to (and in the
direction of) the velocity of the charge. Thus, even
if the charge had no mass otherwise, it would have
a mass associated with its electromagnetic field.
Two other experimental observations are very important when considering the origin of the mass
of the electron. We know that if you accelerate
a charged particle it radiates. This energy it radiates must be supplied by whatever is accelerating the particle. Hence it takes much more energy
to accelerate a charged particle than it would an
uncharged particle (assuming you could figure out
how to accelerate an uncharged particle!). This
problem is more severe the smaller the mass of
the particle. Thus, electrons are generally accelerated in linear accelerators (since then you don’t
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have to worry about the acceleration and radiation
associated with bending the path of the electron).
Finally, scattering experiments with electrons have
failed to identify any finite electron radius, it is as
though they really are point particles.
These experimental observations are difficult to
reconcile either classically or using quantum theory. In fact, in spite of the great success of relativistic quantum field theory, there is still no adequate or widely accepted explanation of the electron’s mass (or its self energy). The situation with
other particles is similar. For charged particles it
is clear that some of the mass is electromagnetic in
origin, but how much, and what accounts for the
rest of the mass is unclear. I should point out that
there exist related elementary particles, the muon
and the tauon, which are like electrons except that
they have much larger mass and a shorter lifetime
(for example the muon has a mass 207 times that
of the electron and a lifetime of about a microsecond whereas the electron is stable). Clearly there
is something missing.
Back to the classical electron model: We also
know that while the electron has a charge, it has no
electric dipole moment (or any higher order electric
Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19
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moment). Thus, the electron’s charge distribution
is spherically symmetric. Interestingly enough, the
electron has a magnetic dipole moment (but no
higher magnetic moments). The classical explanation for this is that the electron is rotating at some
constant rate. This rotation gives rise to an electric
current at the surface of the electron which results
in a dipole moment (you can calculate this next
semester). It may seem curious that all electrons
rotate with exactly the same angular velocity. The
quantum mechanical reconciliation of this with the
notion that the electron is also a point particle is
that such particles possess an intrinsic quantum
mechanical property called “spin”. The spin of the
electron is 1/2.
In an electron-positron annihilation experiment
we have
e+ + e− → 2hν
Two photons are required in order to conserve linear momentum in this process. The minimum energy of the two photons is 2m0c2 where m0 is the
rest mass of the electron or positron.
The electromagnetic self-energy of a positron must
be identical to that of an electron because the in-
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tegral involves E 2. If
ZZZ
1
E 2 dτ = m0c2
2
T
Then, assuming that the electron is a uniformly
charged spherical shell having a radius R, we would
calculate that the self-energy is
e2
= m0 c2
We− =
8π0R
which gives, for the classical radius of the electron
(1.6 × 10−19)2
e2
R=
=
8π0m0c2 8π · 8.85 × 10−12 · 0.91 × 10−30(3 × 108)2
which reduces to
R = 1.405 × 10−15 m
This is the “classical radius of the electron” (Note:
depending on the choice of charge distribution..a
spherical shell, a solid sphere, some radial dependence on charge distribution..you will come up with
slightly different values for the radius).
The size seems reasonable so this is a rather attractive picture. However, one important question
to consider is: What holds the electron together?
To treat any further the basic questions which
have arisen we need to go beyond classical electromagnetic theory.
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1) we need relativity whenever the energies involved approach m0c2
2) we need quantum theory for such small distances and to consider individual particles like
electrons
3) relativistic quantum electrodynamics has resolved many of the technical problems which
have arisen and has increased our understanding of the nature of these particles,but it hasn’t
explained the most fundamental questions regarding the origin of a particle’s mass and the
nature of electric charge.
Electrostatics and Conductors
A perfect conductor contains an unlimited supply of completely free charges.
Real conductors approach this ideal if you don’t
look too closely. For example, a metal like copper
is an excellent conductor but it only has roughly
one free electron per atom. Of course that is a
lot of electrons, so it can support very large currents (but not infinite). As the electrons in a metal
move around they occasionally collide with impurities or are scattered by “lattice vibrations” which
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occur at finite temperatures. These sorts of processes result in a finite“resistance”to electrical current and some associated heat dissipation (that’s
why a toaster works). Even a superconductor such
as N bSn is not a perfect conductor. While it is
true enough that it has zero resistance at temperatures below it’s superconducting transition, it cannot support an infinite current.
We can make a few immediate observations concerning electrostatic fields in perfect conductors:
~ = 0 inside a perfect conductor. If we place
a) E
a conducting object into a region where there
~
is a static electric field E
the free charges within the conductor rearrange
themselves in such a way that the electric field
inside exactly cancels the external field. If there
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remained any field inside, the free charges would
move in response to the force set up by that
field. In the steady state the field inside a perfect conductor must be precisely zero. Thus,
a perfect conductor is also a perfect dielectric.
(We’ll discuss dielectrics later.)
b) % = 0 in the interior of a conductor. The differential form of Gauss’ Law states that
~ ·E
~ = %
∇
0
~ = 0 then % = 0.
Clearly if E
c) As a result of b), any net charge or any unbalanced charge must be on the surface.
d) The potential V is constant everywhere on and
inside a conductor (if it weren’t a current would
flow distributing charge from one part of the
conductor to another...in the steady state the
charge distribution must be such that there are
no potential differences anywhere in the conductor).
~ is perpendicular to the
e) The electric field E
surface of a conductor. Since the surface is
an equipotential (by d) above) then, for ~r =
[x, y, z] on the surface, V (~r) = constant. We
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saw previously that for a surface defined by the
expression ϕ(x, y, z) = constant, the gradi~ is normal to the surface. Thus, E
~ =
ent ∇ϕ
~ (~r) is ⊥ surface.
−∇V
A Conducting Sphere between Two
Charged Plates:
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