LECTURE NOTES FOR PHYSICS 308: ELECTRICITY AND MAGNETISM, Spring 2007
TEXT: GRIFFITHS' INTRODUCTION TO ELECTRODYNAMICS
Last update: 3/30/2007
SOME USEFUL REFERENCE STUFF:
Greek alphabet, metric prefixes, conversion factors
ASSIGNMENTS: (Subject to change: check back often.)
Due Friday, April 27: HW #10:
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.)
Due Friday, May 4: HW #12: Fresnel handout: Questions 3, 4.
"Dailies":
GO TO LECTURE
25,
26
LECTURE 25:
9.3.1: ELECTROMAGNETIC WAVES IN MATTER
Now for the practical stuff. A few sections go we developed equations for electromagnetic waves in free space, where there are neither charges nor current. And yet a lot of optics deals with the motion of electromagnetic waves through matter such as glass or water. (Optics would be pretty dull if we only considered light travelling in a vacuum.) How do we cope?
We can rewrite Maxwell's equations in terms of the two vector fields E and B, and two new fields D and H. If we do this Maxwell's equations in matter appear similar to the equations in free space, only with μ0 being replaced by μ and ε0 being replaced by ε. (The latter is just a change by the factor of the dielectric constant, κ.)
This leads to only two fundamental changes. First of all, the speed of light changes from v =(μ0ε0)-1/2 to v =(με)-1/2.
In the laboratory we describe this by saying that the speed of light slows down by a factor of n, the index of refraction, where n=c/v. The other change is that B0/E0 is changed by the same factor of n:
B0=E0/v=nE0/c.
Cool! Since
μ nearly equals
μ0
for most transparent materials,
n2 = ε: the index of refraction and the dielectric constant (optics and electrodynamics) are very, very closely related.
9.3.2-3: FRESNEL'S EQUATIONS
Now we are finally ready to deduce the laws regarding light traveling between two transparent media. This is done in your book in sections
9.3.2 and 9.3.3, however, I will provide you with a two-page summaries (1) that show the derivation of the resultant equations, Fresnel's equations. Look these over before class.
A few issues come up once we have Fresnel's equations. For one, they produce the interesting (and not surprising) result that, if there is no change in the index of refraction, there is no reflection. This is called 'index matching', and it is very handy to know about. Hopefully your instructor will put together some cool demonstrations of it. If not, be sure to ask him about: (a) Making a pyrex test tube 'disappear' in oil, (b) how a white hair looks under a microscope both with and without immersion in an index-matching solution, and (c) irridescent butterflies. He'll welcome the chance to digress.
Another physical phenomenon which occurs is the Brewster angle,
the angle at which a certain polarization of light passing between two transparent media is totally transmitted with no reflection. We will introduce this phenomenon through the Fresnel equations, but we will also show its origin in a much easier to visualize way. (Maybe next class)
One last very useful rule of thumb about Fresnel's equations. If you solve for the reflection between air and glass (n=1.5), you get r=0.2, or R = (0.2)2 = 4%. That's a useful factoid that comes in handy every once and again.
YSBATS FOR CHAPTER 9:
Memorize:
Relation between c, μ0,
ε0
Relation between v, n, c
The temperature of the blackbody radiation (to the nearest degree K)
Age of the Universe (within factor of 2)
Understand:
How decoupling and recoupling leave us with two independent plane waves
The mathematical, physical origin of polarization
How magnetic charges would alter the Universe as we know it
Physical origins of Fresnel’s eqs.
Index matching, some examples
You should be able to:
Calculate E(t), given B(t), or vice versa
Comfirm that a set of E(t) and B(t) vectors obey Maxwell's equations
Convert between Poynting's S vector (power per area), Erms, Brms, and u.
Keep for future reference:
Maxwell's Equations (general form, magnetic monopole form, fields-in-matter form, and free space form)
Relation between n, μ0, ε0,
μ, and ε
Relation between k, ω, c
Fresnel Equations
Brewster’s angle
Radiation pressure
LECTURE 26:
11.1.2 DIPOLE RADIATION
The only thing I want to cover in Chapter 11 (As fate would have it, there's no time left to cover anything more.) is the material in Section 1.2 on dipole radiation. The rest of the chapter covers other types of sources of radiation, so why do we cover this one? (a) It is the simplest example to study, and would help us should we decide to tackle the other types of sources, and (b) it allows us to understand two interesting physical systems: the linear antenna -- as either an emitter or receiver -- and the individual oscillating atoms and/or molecules that give rise to the Rayleigh scattering that makes the Earth's sky appear blue.
We will cover the derivation in this section of the book. Look over it. The important thing about this derivation is the notion of 'retarded potentials'. These arise because you need to account for the fact that the field one light year away from a dipole, for example, is the result of the dipole's condition one year ago, not its present condition: it takes an electric field time to propagate through space. When we take the perfect dipole approximation limit, this derivation leads to an electromagnetic wave whose power is proportional to the fourth power of the frequency at which the dipole oscillates. Hence the scattering will be much larger for blue visible light than for red.
What the book doesn't supply is the rationale for assuming that molecules in the sky behave as oscillating dipoles. We will show that the mathematics for the electron cloud distribution in a molecule is the same differential equation as for the driven and damped harmonic oscillator: this is a resonance phenomenon. The chief thing to remember here, though, is that we are looking at the behavior far away from resonance. Resonance for these molecules is well into the ultraviolet frequencies. The other thing to remember is that, if the scatterer is close to the same size as the wavelength of what is being scattered, different parts of the molecule will be scattering out of tune with each other and cancel out. This is why water droplets in clouds appear white (wavelength-nonspecific scattering) while the molecules in the air make the sky appear blue (Rayleigh scattering): water droplets in clouds are just too damn large. We will discuss other examples of Rayleigh scattering and non-Rayleigh scattering.
One last item about scattering. Scattered light can be highly polarized. This is the result that polarization is always perpendicular to the direction of motion of light. In fact, this is why the zero reflection at the Brewster angle occurs. Weather cooperating, we will go outside with polarizers and observe the sky and see whether the [hopefully] blue sky is polarized in certain directions. (It should be.) Or, if you wear polarized sunglasses, tilt your head one way or the other as you look at a cloudless sky. In which directions is the sky highly polarized?
YSBATS FOR CHAPTER 11:
Understand:
Origin of Rayleigh scattering (For example, is it magnetic quadrupole?)
What is a retarded potential, its connection to our earlier work
Some examples, counterexamples of Rayleigh scattering
Why large oscillators do not produce Rayleigh scattering
Why scattered light is often polarized
Memorize:
Power dependence of Rayleigh scattering on f (or on ω)