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 #8: Due Friday, April 6: Problems 7: 1, 5.
HW #9: Due Friday, April 13: Purcell handout, 6:34, 35.
HW #11: Due Friday, April 20:
1. Derive the units of Jm and qm from Maxwell's Equations in the presence of magnetic charges.
2. (a) If magnetic charges existed but electric charges did not, determine the source equations for static electric and magnetic fields. (Check out Prob. 7.55B.)
2. (b) Consider the analogues (in this hypothetical Universe) of the Lorentz force Right-Hand Rules: are they Right-Hand Rules, and what do they describe? (The force on what?) (Use the results of Prob. 7.60B.)
"Dailies":
Tuesday, April 10: none
Thursday, April 12: Show that the inductance of a solenoid of radius R is L=μ0n2lπR2.
Tuesday, April 17: Show that the divergence of the curl of a vector function is zero.
GO TO LECTURE
21,
22
LECTURE 21:
LENZ' LAW: ("Nature abhors a change in [magnetic] flux.")
Now what's the direction of this Emf? It turns out there's a simple way to remember how to figure out the direction. As the textbook's author eloquently states, "Nature abhors a change in [magnetic] flux.". What this means is that the Emf that is generated by changing flux will try to keep the flux from changing. I like to think of it as a 'reactionary' force. Like all reactionary forces, this is ineffectual in the end, but that doesn't keep it from trying. The induced Emf produces a current in the loop which creates a magnetic field which tries to keep the flux from changing. What does this mean for you practically? It means you have to keep that right hand ready in order to figure out the direction of the induced Emf. But work backwards. Figure out which way the induced magnetic field would have to point to maintain the flux, then figure out the direction that current would have to flow to produce that magnetic field, then figure out what kind of Emf would produce that current. Easy as that.
INDUCTANCE: 
, 
The book talks about two types of electromagnetic inductance, mutual inductance and self-inductance. We will focus on self-inductance. This is the physical phenomenon that takes place in those electrical components called inductors which you've played with in lab. Consider a solenoid. If you run current through it, this produces a magnetic field perpendicular to the plane of the coils. This produces a flux through the coils, and therefore the solenoid does not like you changing the current through it, because that will change the magnetic flux through the coils. This means that changing the current induces an Emf in the solenoid, in a direction which tries to keep the current from changing. (Any device in which the coils that have flux in them are the same ones that produce the magnetic field is such a 'self-inductor', known more familiarly as an 'inductor'.)
ENERGY IN MAGNETIC FIELDS: 
, 
(Compare to 
, and 
.)
Remember how energy is stored in electric fields? Time for another electrostatic-magnetostatic analogy. If I turn off the current in the solenoid, this induces an Emf. Where does the energy come from to cause this Emf, and to temporarily power the current to continue to flow? We can think of it as stored in the magnetic field. In the same way, if we have a charge stored in a capacitor, we can short out the capacitor and get current to flow. Where did that energy come from? From either the charge, or the electric field within the plates. (Either point of view is equivalent.) So the energy density in electric field is proportional to E2, and the energy density in a magnetic field is proportional to B2. The energy stored in a capacitor goes as Q2, and the energy stored in an inductor goes as I2
LECTURE 22:
SECTION 7.3: MAXWELL'S EQUATIONS
FIXING UP AN INCONSISTENCY:
There is an inconsistency in the laws of electric and magnetic fields as they now stand. If you take div(curl(
)), you get the time derivative of div(-
), and both sides should be equal to zero, which they are. Notice, as the book points out, and as you are asked to as a daily assignment, that the divergence of the curl of any well-behaved normal mathematical vector function is zero. (div(curl(
))=0)
But now notice that if you take div(
), the left-hand side ought to be zero and the right hand side will be μ0div(
). But
div() is only zero if the charge density, ρ, is constant. Maxwell set out to fix this. And he fixed it by a just adding another term to :
OK, Maxwell's kluge fixes the mathematics, but what does it mean? Notice how this expression looks a lot like this source in Faraday's law, 
. The righthand term in that expression means that a change in magnetic field can produce an electric field which curls around of the magnetic field lines. By analogy, we would expect that if we have an electric field which is varying with time, this will produce a magnetic field lines that curl around the electric field lines. What is really cool about this is that if we have electric fields which are varying sinusoidally with time, they will produce magnetic fields also varying sinusoidally with time, etc. That is, a varying field can produce other varying fields ad infinitum. This will give rise to electromagnetic radiation, which we will look at more closely in Chapter 9. In fact, this was the big breakthrough of Maxwell's work: he predicted the existence of electromagnetic waves. Now this is kind of like J. J. Thomson's discovery of the electron. Edison had already been harnessing electrons for decades in various electric inventions without understanding the nature of the electricity which he was exploiting. Same thing with electromagnetic waves. Optics is all about electromagnetic waves. Visible light, infrared, ultraviolet, radio, and other such waves are electromagnetic waves.
The experimental breakthrough came with Heinrich Hertz, who discovered that he could produce a spark on one end of the room which could be detected on the opposite side of the room. This led to the exploitation of radio waves by Guglielmo Marconi. The really amazing thing about all this is that Maxwell predicted the velocity of these waves, and it is a function of the permittivity and permeability of free space and it is exactly equal to the speed of light which had already been measured centuries earlier by Ole Rømer.
MORE SYMMETRICAL VERSIONS OF MAXWELL'S EQUATIONS:
We can now sum up what we know about electromagnetics with four equations, "Maxwell's equations".
These equations are very nearly symmetric. The symmetry is destroyed by the apparent lack of magnetic charges ("magnetic monopoles") in the Universe (current research status, no-monopole song, more Maxwell Equation music). We can make the symmetry more obvious in either of two ways:
(a) First consider Maxwell's equations in free space (where there are no charges or currents available)
(b) Next consider what Maxwell's equations might look like if magnetic charges existed:
,
where ρm is the density of magnetic charge, and
m is the magnetic current, the rate at which magnetic charge flows. (The units not being symmetrical in the equations above is just a sad consequence of our choice of units: in CGS units we would set
ε0 = μ0 = 1.)
MAXWELL'S EQUATIONS INSIDE MATTER:
Finally, let's consider what happens inside materials. The big difference is that there are bound charges (for electric fields in matter) and bound currents (for magnetic fields in matter). This makes it useful to define two new fields, (They were already defined in Chapters 4 and 6, but we didn't have much use for them until now.)
and 
. If this seems like a real nuisance, bear this in mind: living with these new fields means that we can
replace ρ and with ρf and f, the free charge and current densities. That is, we won't have to worry about the densities of charge or current inside the material of interest, densities that are responses to the fields we impose from outside, and densities that we cannot directly change.
All that we are actually going to do with this final version of Maxwell's Equations is to use it to do some interesting optics. It is this variation of Maxwell's equations that causes much of optics that we have taken for granted up till now: the laws of reflection and of refraction in transparent media. We will see why light bends as it passes between material, and why it reflects at an angle equal to the incident angle, and find out that it is because of electrodynamics. But that will have to wait for Chapter 9.
FINAL REVIEW SINGALONG
YSBATS FOR CHAPTER 7:
Memorize:
Ohm's law (3 different forms)
Faraday's and Lenz' Laws
The Back-emf of an inductor
The energy stored in an inductor, the energy density in a magnetic field
Understand:
The microscopic origins of resistivity
Why Maxwell's equations are asymmetric
How they can be made more symmetric
You should be able to:
Calculate motional Emf
Calculate Emf due to change in flux
Compare the total energy stored in a solenoid, for example, to the energy density of the magnetic field inside
Keep for future reference:
Faraday's law in differential ('microscopic') form
Maxwell's Equations in the four versions above