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Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 1 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. Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 2 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 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 3 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 4 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- Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 5 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. Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 6 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 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 7 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 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 8 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 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 9 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: Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 10 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 11 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 12 Physics 2460 Electricity and Magnetism I, Fall 2006, Lecture 19 13