LECTURE NOTES FOR PHYSICS 308: ELECTRICITY AND MAGNETISM, Spring 2007
TEXT: GRIFFITHS' INTRODUCTION TO ELECTRODYNAMICS
Last update: 1/16/2007
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.)
"Dailies":
Tuesday, April 17: Show that the divergence of the curl of a vector function is zero.
Tuesday, April 24: Verify the identity for the curl of the curl of a vector function. Do this by writing it all out in component form.
GO TO LECTURE
24
LECTURE 23:
EXAM # 3
LECTURE 24:
SECTION 9.2.1: DECOUPLING MAXWELL'S EQUATIONS
All right, we have Maxwell's equations, now what do we do with them? Let's start by calculating curl(curl(
)). After a couple of steps -- including the one that stars in the "Daily" for next time -- we arrive at an expression that shows that each component of the electric field obeys the wave equation. Repeating this procedure for curl(curl(
)), we get a similar result for the magnetic field. What we also get is the result that the velocity of the electric and magnetic waves is the same, v =(μ0ε0)-1/2=3.00x108m/s, the speed of light. Coincidence?
We will go over the mathematics of this in class, which begin with Maxwell's equations in free space:
We will also show that the equations we obtain are consistent with the wave equation, and we will look at a few functional forms that are valid solutions to that equation.
SECTION 9.2.1: RECOUPLING MAXWELL'S EQUATIONS
We have what appear to be six independent wave equations, the x-, y-, and z- components of
and . However, these six are not totally independent of each other. We simplify things (really!) by defining the z-axis to be the direction of propagation of the electromagnetic wave. Then, assuming reasonably arbitrary forms for E and B, we apply the two curl equations from Maxwell's equations, we see how the electric and magnetic fields are related.
The chief result of this exercise, which I will ask you guys to do on the board in groups of two, will be (a) to show that the electric and magnetic fields are transverse to the motion (and result in two independent polarizations), (b) to find the relation between ω, the angular frequency of the wave, and k, the wavevector, and (c) to derive the relationship between the magnitudes of the electric and magnetic fields, namely that B0 = E0/c. (In CGS units, B0 = E0.)
9.2.3: ENERGY AND MOMENTUM
While the expressions for the electric field components of a wave give us all the information we need, they're not the most useful quantities for describing the waves. More useful is the rate at which energy is travelling through space on the crest of that wave. We already have expressions for the energy density in an electric or magnetic field. The first thing we will show is that, given the relationship between B0 and E0, electromagnetic waves contain the same amount of energy in their electric and magnetic fields.
Next we can define a vector which gives the intensity of the electromagnetic wave and tells us the direction in which it is poynting. This is known as the Poynting vector (after John Henry Poynting, 1852-1914). It has units of power per area, exactly the sort of units that you would need in order to determine whether not there's enough light in the room to read your text by. The expression for the Poynting vector is
Notice that, for electromagnetic waves in free space, S=cu, where u is the energy density of the electromagnetic field.
I want to introduce one last expression for a property of the electromagnetic wave: radiation pressure. This is nothing more than the force per area: P=F/A. You may be familiar with science fiction or speculative fiction in which people talk about building enormous solar sails to transport spacecraft. The question is how large a force does like to incident on such a sale exert on it. From relativity we know that p = E/c. Consequently the radiation pressure is P=u, that is, the radiation pressure exactly equals the energy density. Compare this to the Bernoulli principle in classical fluid dynamics.
By the way, this radiation pressure does not explain how a radiometer works, but does explain how a solar sail does.
THE 3K BACKGROUND RADIATION
At this point, we'll take a brief break from our book to talk about the background microwave radiation from the Big Bang, placing it in its scientific and historical perspective. This microwave background radiation is the main proof we have of the Big Bang and the age of the Universe. Check out this site, plus I'll provide a handout that tells a little bit of the history of the 'discovery' of this phenomenon, or rather how the scientists came to understand what it was they were seeing.
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