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.)
Due Friday, March 9 at noon: HW #5: Problems 3.6, 27, 32, plus, solve for V(r,θ) in all space, if V(R,θ) = C cos3θ on the boundary, R of a sphere.
Due Friday, March 16 at noon: HW #6: Problems 4.19 (capacitance only); 5.1, 3.
"Dailies"
Due Tuesday, March 6: For Figure 3.34B, calculate the dipole moment for each of the following choices of origin: the leftmost charge and the topmost charge.
Due Tuesday, March 13: Problem 5.4. (Show that F=Ika2.) What is the direction of the force?

GO TO LECTURE 13, 14

LECTURE 13:
CHAPTER 4: ELECTRIC FIELDS IN MATTER

I will take a completely different approach with this material then the text does. We will begin by looking at what effects these electric fields can produce in macroscopic objects, and then we will consider the microscopic picture of how these effects arise.

DIELECTRIC CONSTANT
Consider a parallel plate capacitor. Often such a capacitor has some nonconducting material between the two plates. This material, called a dielectric, serves three principal purposes: 1. It serves to separate the plates, keeping them from shorting each other out.
2. It prevents the two plates from sparking. Such sparking is known as dielectric breakdown, and occurs at about 3MV/m in dry air and one-tenth of that in moist air.
3. The material increases the capacitance of the capacitor. This chapter is mostly about trying to understand why this happens. We can start empirically (experimentally) by defining a quantity we call the dielectric constant, κ, or, as the book writes it, εR, the relative permittivity. In other words,
where C0 is the capacitance without the dielectric in place. (Notice that the dielectric constant is generally greater than one, just as the index of refraction is always greater than one.) Now, let's consider what happens when we put a dielectric between two parallel plates, from the standpoint of charge, voltage, and electric field. We will assume that we change none of the dimensions of the capacitor as we do this. First of all, since , increasing the capacitance without changing the charge means that we have a smaller voltage between the plates. Next, because V = E d for the uniform electric field between the plates, this means that putting the dielectric in place reduces the electric field between the two plates. Now, because the superposition of two electric fields adds like a vector, we can conclude that placing a dielectric between two capacitor plates has the effect of superimposing a somewhat smaller electric field, E, which points opposite the field produced by the charges on the plates.

THE DIELECTRIC SLAB
Let us picture a slab of dielectric material placed between two capacitor plates, nearly filling all the volume in between. Draw yourself a picture of this right now, a side view. Mark one of the plates as having positive charges on it, and the other plate as having negative charges. Now draw the electric field (outside the slab) pointing from the positively charged plate to the negatively charged plate. Next, draw the internal, reversed electric field inside the dielectric pointing in the opposite direction. Finally, label the edge of this slab from which this arrow originates as having a positive charge, and the other side with negative charge. What you should find at is that the side of the slab closest to the positively charged capacitor plate is loaded with negative charge, and the side closest to the negatively charged plate is loaded with positive charge.

What is the dielectric? Well, one thing it is not is a conductor. This means that these excess charges on the two faces of the slab do not come about from free charges flowing through the slab. Rather, they come about from charge separation within the atoms and molecules which make up the dielectric. We will sketch in class how a collection of fixed atoms can give rise to such a macroscopic separation of charge.

POLARIZATION
In order for this macroscopic separation of charge to happen, we do need to have microscopic charge separation within the atom or molecule. The process by which the separation occurs on a microscopic level is called polarization. "Polarization" is also the name of a specific physical quantity, the average dipole moment per volume, which is given as
There are two ways by which you can get polarization. Some molecules have a permanent dipole moment: water, for example. The electrons in the hydrogen atoms of water spend more time near the oxygen atom then they do at home with the hydrogen atoms, so this produces a permanent charge separation. On the other hand, in symmetric molecules such as N2, O2, H2, there is no permanent dipole moment, but a temporary dipole moment can be induced. In the presence of an electric field, the electric field distorts the electron cloud surrounding the molecule in such a way that the average position of the electron is separated from the average position of the nuclei. The greater the electric field, the greater the separation. For sufficiently small electric fields, the average dipole moment is usually linear in the electric field for both permanent and induced dipole moments:
where α is the atomic polarizability. In Example 4.1, one sees that the atomic polarizability is of the order of ε0 times the physical volume of the individual atom. This is approximately true throughout periodic table.

As for permanent dipoles in the presence of an electric field, random thermal collisions tend to counteract the effect of the electric field to align this permanent dipole, and so the average dipole moment increases with electric field. This means that we can use the last equation (p=αE) for permanent dipoles too.

PERMANENT DIPOLES
In addition to the conceptual model above that shows how a dielectric can increase the capacitance of a pair of plates, we will also use this chapter as an opportunity to look at electric dipoles in general. There are a number of reasons for this. First of all, in materials that have no net electric charge, dipoles play a very important role, as we've just seen. The second reason is that for magnetic fields, there is no analogue to the electric charge. All magnetic fields are produced by dipoles. As it happens, the mathematics for magnetic dipoles is exactly the same as for electric dipoles, so maybe this is a good place to get our feet wet.

We can look at dipoles in electric fields from either the force approach or the energy approach. The net force on a dipole in a uniform electric field is zero, since the forces on the positive and negative ends of the dipole are identical. If the field is nonuniform, on the other hand, there will be a net nonzero force, but we will get to this later. What a dipole will experience in a uniform electric field is a torque. This is because, although the forces on the two ends of the dipole are equal in magnitude, they point in different directions, and thus act to twist the dipole. The torque on a dipole is given by
In the energy approach, we can calculate the potential energy of a dipole sitting in electric field as being equal to
Now, for the force in a nonuniform electric field. Since the force is the negative gradient of the potential energy --- --- the force on a dipole in a non uniform field is
In real situations, this is a rather messy equation to try to apply. The important thing I want you to get from this is that dipoles have net forces on them only in nonuniform fields.

Digression time. We don't see a lot of electric dipoles, so the story I'm going to tell here is the story of magnetic dipoles. If you have two refrigerator magnets, and you bring them close together, obviously they attract each other. The only way this can happen is if one of them produces a magnetic field which is nonuniform. If we put a collection of refrigerator magnets in a room that was filled with a large, but uniform magnetic field, all that would happen is that the magnets might twist to align themselves with the field, but they would not go flying through space. It is only because individual magnets produce wildly nonuniform magnetic fields that we get this effect that we are so familiar with, namely magnetic dipoles exerting net nonzero forces on each other.

SECTION 2
Skip most everything from Section 2 on, except for Example 4.6, Problems 4.18-4.21, and Section 4.4.3.

ENERGY IN ELECTRIC FIELDS IN MATTER
We saw in an earlier chapter that we can associate an energy density with electric field: . What is the energy inside a dielectric between two parallel plate capacitor plates? Do we use the field outside the dielectric slab, or do we use the field inside? The answer, which will not derive, is a bit of a compromise:
, where u0 is the energy density without the dielectric. Now, since κ is greater than one, this means that more energy is stored in the field when we add the dielectric, even though the electric field inside the dielectric is smaller than it was in that space before we inserted it. Why is this? Well, it takes a lot of energy to separate all those charges, or if the dielectric consists of permanent dipoles, to maintain the dipoles aligned in the face of all the thermal agitation which would tend to randomize their orientations.
YSBATS FOR CHAPTER 4:
Memorize:
Relation between vectors p and P
Expression for how adding a dielectric alters the capacitance

Understand:
How the surface charges on the dielectric arise
The two types of polarizability, the origin of each

You should be able to:
Calculate the impact of a dielectric on Q, C, and V

Keep for future reference:
How a dielectric alters u, U
LECTURE 14:
CHAPTER 5: MAGNETOSTATICS

I have yet to find a textbook that treats magnetostatics well. There's a lot of difficult, confusing material which is usually poorly organized. I will try to help out here by providing an alternate way of organizing this material in these lecture notes. Here is what I will do in these lecture notes for this chapter: first I will present the material on magnetostatics which appears in the book, organized in terms of "forces" and "sources" (nice catchy aid to remembering the two categories). In other words, for most practical problems, what you really need to know is (a) how to calculate the magnetic fields caused by a certain arrangements of currents and (b) the magnetic force on electric objects. Second, I will look at the properties of the magnetic field and current density field, using some vector calculus. Thirdly, I will rehash all of this material in order to point out the similarities and differences between the electrostatic and magnetostatic fields. Finally, I will describe two interesting magnetic experiments that are not in this text.

Conceptually, what do I want you to get out of this material? First, I want you to have a practical handle on calculating magnetic fields and the effects that they have on charges and currents. Secondly, I want you to understand the similarities and differences between electrostatics and magnetostatics fields. More on all of this in the YSBATs.

BACKGROUND
Experimentally, we find that a moving charge located in a region of space containing electric and/or magnetic fields experiences a net force called the Lorentz force:
Notice the first striking difference between the electrostatics and magnetostatics: magnetic force is not parallel to the magnetic field, but perpendicular to it. In order to talk about what causes these magnetic fields, we have to do a little bit of housekeeping here, introducing a a quantity that we should have introduced earlier, an electrodynamic quantity which you probably already recognize as the current, I, the rate at which charge passes past some point in space:
The units of current are Amperes or amps, A=C/s. By the way, the SI units of magnetic field are called Teslas: T=N/(A . m). It is a useful exercise to confirm to yourself that this is the combination of units that would be consistent with the expression above for the Lorentz force. We will probably do similar calculations like this in class, just to practice the algebra of checking for units.

MAGNETIC FORCES
We will introduce two important expressions for (a) on the one hand, the force on a moving charge, and (b) on the other hand the force on a current-carrying wire. In both cases, you need to memorize two things: (a) the vector equation for the force and (b) the "Right Hand Rule" for determining the direction of the force. When we are done with these two expressions for forces, we will look at a couple of the expressions for magnetic sources. Two of these expressions will also have associated Right Hand Rules.

The first expression is for the force on a moving charge. You have actually already seen this, because it is the second term of in the equation above for the Lorentz force. The magnetic force on a moving electric charge is
Now consider a wire which is carrying electric current. We can picture this as containing a stream of electric charges travelling down the length of the wire with some velocity, v. There ought to be some force acting on these charges. One can show that the equation for the force on a moving charge yields this expression for the force on a current-carrying wire:
Now for the "Right Hand Rules". Here the order in which we write the terms on the right hand side is very important. Since these are cross products, we simply use the right hand rules for a cross product. There are a number of ways of stating the Right Hand Rule, differing from textbook to textbook and instructor to instructor. I will give you my version, but you should feel free to come up with your own expression, as long as it is consistent with mine. If we have a vector = x , you can find the direction of by a pointing your fingers in the direction of and then slapping them in the direction of . Now, if you put your fingers pointing in the direction of , your palm will be facing in any one of a variety of directions. But there is only way you can point your palm so as to be able to slap in the direction of without straining your wrist (i.e. without slapping more than 90o). So, after you point our fingers toward , you have to orient your palm in such a way that you can slap in the direction of . Once you have done all of that, you find that your thumb is pointing in the direction of the vector .

To recap, to find the direction of the force on a moving charge, point your fingers in the directions of QV, and slap in the direction of the magnetic field, . to find the direction of force on a current-carrying wire, point your fingers in the direction of the current and then slap in the direction of . In either case your right thumb points in the direction of the force. Any of you who are left-handed will have an easier time because you can continue to take notes while your right hand is figuring out the direction of force.

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