LECTURE NOTES FOR PHYSICS 308: ELECTRICITY AND MAGNETISM, Spring 2007
TEXT: GRIFFITHS' INTRODUCTION TO ELECTRODYNAMICS
Last updated: 1/16/2007

SOME USEFUL REFERENCE STUFF: Greek alphabet, metric prefixes, conversion factors

ASSIGNMENTS: (Subject to change: check back often.)
HW #7: Due Friday, March 30: Problems 5: 8, 10A, 14.
HW #8: Due Friday, April 6: Problems 7: 1, 5.
No new "Dailies":
GO TO LECTURE 17, 18
LECTURE 17:
SECTION 7.1: ELECTROMOTIVE FORCE
OHM'S LAW:             V=IR,             ,            
When current flows in an object, the voltage drop along the length of the object is usually proportional to the current. This is an experimental fact for many materials in specific ranges. It doesn't work for diodes, or for objects like tungsten filaments for which the energy dissipated by the current changes the temperature of the object.

This macroscopic equation is like Gauss' Law and Ampere's Law: it can be replaced by an equation that holds (or doesn't) at a point in space. The pointlike analog of current, I, is current density, , and the pointlike analog of voltage is electric field. The pointlike analog of resistance, R, is resistivity, ρ. So the pointlike versions of Ohm's Law are the ones given above. Notice that resistivity and conductivity, σ, are inverses of each other: ρ=1/σ

These microscopic versions of Ohm's Law are not strictly true, even for 'Ohmic' materials. They are true if two conditions hold, namely, first there is no magnetic field, and secondly the material is isotropic. Multicrystalline materials, such as the metal you would find in a wire, are isotropic. But for single crystals, we have to be careful. In a more complex model, the conductivity and the resistivity are not simple scalars, but matrices. In fact, the Hall effect, which produces an electric field perpendicular to the flow of current, can be thought of as an off-diagonal element in the resistivity matrix.

The book makes another point which I mentioned several chapters ago. Part of my motivation for showing you novel ways of solving Laplace's Equation was my assertion that you can use it to solve electric current flow problems. The book shows that if you start with the continuity equation, and consider only steady currents, and assume constant conductivity, you get back , and Laplace's Equation, . I just mention this to justify what we did months ago.

DRIFT VELOCITY:             v_d
What is the microscopic effect of putting a voltage across a conducting material? The voltage sets up an electric field in the material. This E-field produces a force on the charge carriers. You would thus expect them to be accelerated down the wire. (You would be wrong.) This would be the case if it weren't for microscopic scatterers in material. We can analyze what happens in a conducting material using a semiclassical model known as the Drude model. A single charge carrier accelerates down the wire -- as a result of the applied voltage -- but then scatters in a random direction after hitting some scatterer in the material. There are two interesting things about this model. First of all, it suggests that if we could take out all those scatterers, we would not have to provide a voltage across our wire to get current to flow. It turns out that there are materials in which this is the case. They are called superconductors. Another interesting thing is that we can calculate the velocity associated with the flow of current. This is called the drift velocity, v_d , and it is superimposed on the thermal motion of the charge carriers.

The average thermal velocity of the charge carriers is zero, but that doesn't mean that they're not moving. In fact, because the atoms have an average kinetic energy associated with its temperature, we would expect that the average magnitude of that thermal velocity would be nonzero. In fact, the drift velocity is much smaller than the average magnitude of the thermal velocity. It's only because the various thermal velocities of the charge carriers average to zero (because they are vectors pointing in random directions) that the small drift velocities of the various charge carriers (which don't point randomly, and therefore don't average to zero) produce a net flow of charge, called current, in the wire. Oh, one last thing. The Drude model is called a semiclassical model only because electrons were not discovered until 1897. We can thus think of electrons as quantized charge, which is what makes them unclassical.

JOULE HEATING: One of the reasons that the microscopic picture of conduction is useful is that it explains why it requires work to move charge in any material other than a superconductor. Because thermal agitation randomizes the direction of the charge carrriers from the direction that the applied voltage is pulling them, this randomizing behaves like a friction. The rate at which energy is lost through this agitation (and any agitation caused by the randomness of scatterers) is given by the expression for Joule heating that you've already learned in Phys 152:
P = V I = V² / R = I² R
This is the macroscopic form for the heating in a charge-carrying material. Notice that if you are given any two of the four following quantities -- I, V, R, P -- you can solve for the other two using Ohm's law and this expression.


EXAM II

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YSBATS FOR CHAPTER 6 OF GRIFFITHS:

Memorize:
The expression for the Lorentz force
Definition of current
Units of B, I
The divergence and curl of the magnetic field
Ampere's law
The four Right Hand Rules: two forces, two sources
Div and curl of B

You should be able to:
Calculate the B-field due to a combination of sources (three long wires or a wire and a coil, etc.)
Find an appropriate Amperian loop for symmetric problems
Apply Ampere's law to [4] simple cases
Calculate Hall coefficient, Hall voltge

Keep for future reference:
The formulas for magnetic field from a long wire, coil, solenoid