LECTURE NOTES FOR PHYSICS 308: ELECTRICITY AND MAGNETISM, Spring 2007
TEXT: GRIFFITHS' INTRODUCTION TO ELECTRODYNAMICS
Last updated: 3/8/2007

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

ASSIGNMENTS: (Subject to change: check back often.)
Due Friday, March 16: HW #6: Problems 4.19 (capacitance only); 5.1, 3.
Due Friday, March 30: HW #7: Griffiths 5.8, 10a, 14
"Dailies"
Due Thursday, March 16: Find the distance between two single coils such that the first two derivatives of B with respect to z will be zero. (This is the Helmholtz coil. Your instructor will give you the first and second derivatives of B, for a single coil, with respect to distance along the axis.)

GO TO LECTURE 15, 16

LECTURE 15:
MAGNETIC SOURCES
The two principal sources of magnetic fields we will discuss are both current carrying wires. One is simply a straight length of wire; the other is a circular loop. At a distance r from the axis of the straight wire -- as long as r is much less than the length of the wire -- the magnetic field equals



where μ₀ is a constant known as the "permeability of free space" which is equal to μ₀=4π   x 10^-7N/A². It is a useful exercise to confirm that the units on the lefthand and righthand sides of the equation equal.

Notice that this is not a vector equation. This means that we need some way of determining the direction of the field. We will introduce the appropriate Right Hand Rule soon.

Straight long wires are not very useful if you're trying to design your own electromagnet. The magnetic field extends into all of space, and falls off very slowly. But most importantly, you would need a very strong current in order to get any sort of decent magnetic field anywhere except right next to the wire. A much more effective source for the magnetic field is a circular coil. The equation for the field near a circular coil is generally messier than for the field near a long straight wire. However, I will give you an expression for the magnetic field at the center of the coil, and then, for your own future reference, an expression for the magnetic field along the axis of the coil. For a coil of radius R, the magnetic field in the center equals



For that same coil, the magnetic field along its axis, at a distance z from the plane of the coil, is given by



Now a single coil is not a very good electromagnet either. The field drops off quickly along its axis. There are two common ways of getting around this. One is called a solenoid. In this, we wrap wires around a cylindrical form, so that we are simply adding a large number of single coils together, each of them displaced along the z-axis from each other. If the solenoid is sufficiently long, then the magnetic field inside the solenoid can be made nearly constant, equal to



where n = the number of coils per length of the solenoid, n=N/L, where N = the total number of coils in the solenoid, and L = the total length of the solenoid. In order to approach this limit, you need to fulfill the following conditions: the solenoid must be much longer than it is wide, and you must be measuring the magnetic field close to the center of the solenoid.

Another way to improve single coils is to put two of them (each of radius R) a distance R from each other along the same z axis. This arrangement is called a Helmholtz coil. This is particularly useful if you don't need a very large magnetic field, but you need the field to be relatively uniform in the volume surrounding the center of the coils. We'll explore this uniformity in a "Daily" problem.

RIGHT HAND RULES FOR MAGNETIC SOURCES
The Right Hand Rules for these two sources --- the long straight wire and the single coil --- are similar to each other but quite different from the right hand rules we developed for magnetic forces. Those right hand rules involved slapping; these right hand rules involve holding your right hand with the fingers curled and the thumb pointing straight out. (more like hitchhiking than slapping) The only two quantities involved are the current and the magnetic field. So these Right Hand Rules are fairly easy to remember. If the current is pointing in a straight line, point your thumb in that direction and the fingers will curl in the direction that the magnetic field curls around that wire. If the current is pointing in a loop, curl your fingers in the direction in which the current curls, and your thumb will point in the direction of the magnetic field inside that loop.

The hardest part is wrapping your brain around the idea that the B-field surrounding a long, current-carrying wire points in a circle. So let's consider a simple example. Imagine a power line, many feet above the ground, in which the current points east. On the ground, the B-field points north. Above the wire, the B-field point south. At the height of the wire, but south of the wire, the B-field points toward the ground. And at the height of the wire, but north of the wire, the B-field points straight up. In each case, in order to figure out the direction of the B-field, you begin by pointing your thumb in the direction that the current points, namely East, and then putting your fingers in the direction at which you want to measure the direction of the field. If you want to know the direction of B on the ground, for example, point your thumb towards the east with your palm facing north, so that your fingers are below your thumb, and you will see that they are curling towards the north. Work through this example to see whether or not you get it. If not, come to class with a bunchload of questions.

THE LAW OF BIOT AND SAVART
It will seem that we have skirted the real, basic issue here: what is the magnetic analogue of Coulomb's law? The reason we have skirted this is that the equation is much uglier than Coulomb's law. The reason for this is that the source of any magnetostatic fields is a current, and there is no such thing as a "point current", as there is a "point charge". This suggests that in order to calculate a magnetic field, we will always need to integrate something. The other problem is that the magnetic fields created by a current is always perpendicular to the direction of that current. As a result the equivalent of Coulomb's law for magnetostatics, the Law of Biot and Savart, has the following, ugly form:



Here I have given you the differential form, unlike the integral which the book gives you, because frankly this is a bit less ugly.


LECTURE 16:
AMPÈRE'S LAW
the law of Biot and Savart is even worse than it appears. Notice that the curly r in the denominator, as well as the curly r unit vector in the numerator both vary as you integrate. If only there were a shortcut like Gauss' law gives us. Fortunately there is --- Ampère's law:



This integral is a closed line integral. That means that you integrate over a path which is a closed path --- a path that returns to its origin. Now I_enclosed is the total current which passes through a surface whose edges are defined by the closed path. For example, consider a wire of circular cross section. If the path of integration were a circle going around the circumference of the wire, then I_enclosed would just be the current in that wire. As a practice, confirm to yourself that Ampère's law is true for the magnetic field produced by a long straight wire. The trick in applying Ampere's law, just like the trick for applying Gauss' law, is to choose an appropriate integration surface. For this wire, the obvious choice is a circle centered on the axis of the wire itself.

DIFFERENTIAL FORM OF AMPÈRE'S LAW
In the same way that we can use Gauss' theorem to come up with a differential form of Gauss' law, we can use Stokes' theorem to come up with a differential form of Ampère's Law:


Unfortunately, since the righthand side is not zero, we cannot define a scalar magnetic potential.

CURRENT DENSITY,
Remember how we had three fundamental quantities in electrostatics --- ρ, , V? We can define three analogous quantities for magnetostatics. Obviously, is the analogue for . In this class, we will ignore the magnetic potential, partly because it is, alas, a vector potential. But what is the analogue for the charge density? Well, since all electrostatics fields are produced by charges, and all magnetostatic fields are produced by currents, there ought to be some kind of "current density". The difference between it and the charge density how, however, is that the current density is a density of current per area, rather than per unit volume. We use the vector to describe this current density. The relation between the current and charge can be summed up, in differential form, as what is known as the Continuity Equation:



where ρ is the charge density. In steady state, the righthand side vanishes: div = 0.

DIVERGENCE OF
We saw above that Ampère's Law in differential form gives us an expression for the curl of . It turns out that the divergence of the magnetostatic field is zero.



Now, remembering that the divergence of the electrostatic field was proportional to the charge density, we can interpret this expression as telling us that there is no magnetic equivalence of a point charge -- which would correspond to a "point current". Such an equivalent, which we would call a "magnetic monopole", simply does not appear to exist in our universe. As it happens, there are a number of theorists who have come up with theories which demand the existence of magnetic monopoles. That's the good news as far as the existence of magnetic monopoles goes. The bad news is that most of these theories posit the existence of exactly one such magnetic monopoles in our Universe. How does this affect you personally? Well, the odds of you ever running into that one hypothetical magnetic monopole in the course of your brief existence in this out of the way corner of the universe is practically nil. So, even if div is not strictly equal to zero, for your purposes it is convenient to pretend that it is.

COMPARE AND CONTRAST ELECTROSTATICS AND MAGNETOSTATICS
All right, let's began with a comparison between electrostatics and magnetostatics. First of all, electric charges produce electric fields. Electric currents produce magnetic fields.

COULOMB, BIOT AND SAVART: ,
Second, Coulomb's law describes how a point charge produces an electric field. The law of Biot and Savart describes how a point current produces a magnetic field. The problem with using the law of Biot and Savart is twofold: first of all, it involves a cross product, that is, unlike Coulomb's law, the magnetic field is not parallel to the position vector. Second of all, there is no such thing as a point current. It is necessary to integrate over the entire current loop in order to calculate the total magnetic field.

LORENTZ FORCE:
Next, the force produced by a magnetic field is not parallel to the field itself, as is the case for a electric forces and fields. Again, there is a cross product, which complicates the mathematics.

GAUSS and AMPERE: ,
Now, since Coulomb's law and the law of Biot and Savart are both "formal definitions" (i.e. practically worthless), we need to use something else to solve for the field. For electrostatics, we use Gauss' Law, for magnetostatics, we use Ampere's Law. In Gauss' Law, the left-hand side is a closed surface integral, with a dot product between the field and the increment. In Ampere's Law, the left-hand side is a closed line integral, with a cross product between the field and the increment. In Gauss' Law, the right hand side is proportional to the enclosed charge. In Ampere's Law, the right hand side is proportional to the enclosed current. Notice, that the electric field is related to charge and the magnetic field is related to current.

DIFFERENTIAL FORMS OF GAUSS AND AMPERE: ,
Let's compare the differential forms of Gauss and Ampere. The differential form of Gauss' Law relates the divergence of the electric field to the charge density. The differential form of Ampere's Law relates the curl of the magnetic field to the local current density. Again, electric = charge, magnetic = current.

CURL E, DIV B: ,
We looked at two of the vector differentials of the two types of fields. Let's look at the other two vector differentials. We saw that the curl of the electric field is zero. This means that the electric force is conservative. Now we see that the divergence of the magnetic field is zero. This means that there are no magnetic monopoles.

These four vector equations for the electric and magnetic field give us what we will call Maxwell's equations, which we will need to fix up a little in a later chapter.

LAW OF CONTINUITY:
OK, now for some new stuff. First of all, electric and magnetic fields are related because their sources, the electric charge and electric current, are related. And this is shown in the Law of Continuity.

---

YSBATS FOR CHAPTER 5:

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

Understand:
What Rowland's experiment shows
What's useful about the Hall effect

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