University of Maryland, College Park

Spring 2010, Professor: Ted Jacobson

Notes, Demos and Supplements

In these notes I'll try to mention things I talk about in class that are not also in the textbook,

as well as supplementary material.

Please do not assume that these notes are even roughly complete.

Tuesday May 11 review for exam

One thing I mentioned during the review, that we didn't cover in class, wis

frustrated total internal reflection.

And a student asked why it's called a "gauge transformation".

I explained its origin with Hermann Weyl's failed unified field theory of electromagnetism

and gravity, in which the (4-dimensional) vector potential determined the change of scale of an object as

it moves areound in spacetime. I said the theory was later reinterpreted in terms of the phase of

the quantum wavefunction, rather than the scale of the object, by Dirac. But I looked it up, and

it seems that it was Weyl himself, and also Fock, and London, who did this first.

Monday May 10

relativity in spacetime diagrams:

A paper explaining more or less what I did in class is

Spacetime and Euclidean Geometry, by Dieter Brill and Ted Jacobson

The lightlike line segments are limits of timelike ones as well as spacelike ones.

On these grounds I argued that the proper time of the timelike lines and the

proper length of the spacelike lines must go to zero as the lines approach their

common limit, the lightike line.

Proof that square of (c x proper time) is twice the Euclidean area of the light rectangle.

Used two inertial observers, Alice and Bob, who exchange light signals after equal

proper times for each of them. The light triangle (half the rectangle) areas were compared

by writing each area as 1/2 base times height. The ratio of the bases was seen to be

the reciprocal of the ratio of the heights, so the areas are the same. This shows that the

square of the proper time is proportional to the area. The proportionality constant can

be evaluated by a simple special case, the observer who bisects the 90 degree angle

between the light rays. Note that these pictures break the symmetry: there appears to

be a preferred "vertical" observer. But the spacetime structure is independent of this

diagram feature.

The time^2 = area easily demonstrates time dilation and the twin effect.

The twin effect is analogous to the fact, in Euclidean geometry, that not all

paths connecting two points have the same length.

The spacetime Pythagoras theorem expresses the proper time of a timelike displacement

in terms of the time and length measurements of a given observer:

(proper time)^2 = t^2 - (x/c)^2

I proved the theorem using the time^2 = area relation.

Thursday May 6

- runaway and pre-acceleration solutions of Abraham-Lorentz equation

- pertubative method to reduce the order of the equation: series solution in powers of tau

- origin of self force: I didn't really go into this. But I mentioned Griffiths' dumbbell model

of a charge q as a pair of charges q/2 separated by a distance d that he takes to zero at the end.

The force of one end on the other gives the "interaction" part of the self-force F_int(q), which

Griffiths points out is half of the total. To get the other half, in problem 11.20 Griffiths changes

the model to a continuous charge distribution. But instead we can argue like this:

Let F(q) be the TOTAL d-independent part of the self force on a charge q in the limit as d goes to zero.

It breaks up into the force of the two q/2's on each other, plus the force of each q/2 on itself:

F(q) = F_int(q) + 2 F(q/2).

The function F(q) must be proportional to q^2, since the field is proportional to q and the force

to q times the field. Hence F(q/2) = F(q)/4, so the above equation implies F(q) = 2 F_int(q). QED.

- Relativity: Maxwell's equations imply electromagnetic waves which travel at the speed of light.

19th century physicists believed this speed was measured relative to the rest frame of the aether.

This was strange however because Newtonian mechanics has no preferred rest frame (it has

"Galilean symmetry"). Moreover, Maxwell's theory had an obvious element of relativity of motion:

electromagnetic induction works the same whether it is the magnet that moves or the wire that moves,

which strongly suggested (to Einstein, who began his first relativity paper commenting on

exactly this) that in fact Maxwell's theory has no preferred rest frame, and in particular that the

aether frame is undetectable. Theoretical work by Lorentz and Poincare showed this to be the case,

and the Michelson-Morely experiment demonstrated that light propagates at the same speed in

all directions relative to the (moving) earth. So the conclusion was that there was in fact no preferred

frame...However, this was incompatible with the usual Galilean relation between relatively moving

observers, who should see waves as traveling at different speeds. Einstein's resolution of this conundrum

was to physicalize the process by which coordinate systems are defined, using light rays to synchronize

clocks. This produced relations between coordinates that were identical to those which Lorentz had

previously shown to be symmetries of Maxwell's equations. But what for Lorentz was just a mathematical

symmetry, was for Einstein a relation between different physical observations.

To start things off, I took the Lorentz point of view and just asked for which changes of coordinates

(t,x,y,z) -> (t',x',y',z') is the wave operator unchanged, box = box'? Because I want to show you something else,

we don't have time to show that the entire set of Maxwell equations has this symmetry, when in addition the

fields are transformed appropriately. However it should perhaps at least be plausible, since the retarded

potentials are exactly solutions to the wave equation in Lorenz gauge.

The primed observer moves at speed v relative to the unprimed one in the x direction, hence x' = 0 = y = z

corresponds to the line x = vt, y = z = 0. That is x' = 0 is x - vt = 0. Thus the most general linear transformation

(I won't try to motivate the linearity here) that could relate the coordinates is (with a convenient parametrization)

t' = g(at + b/c

x' = g(x - vt)

where g, a, and b are some constants. We expressed the partial derivatives wrt t and x in terms of the partial

derivates wrt t' and x', and required d

This is the Lorentz transformations.

Rather than going further down this algebraic path, I want to show you a coordinate independent, geometrical

way of understanding special relativity...

Simultaneity: The Einstein radar convention for establishing simultaneity yields different definitions of

simultaneity for inertial obesrvers in relative motion. I drew a spacetime diagram illustrating this.

Proper time: I claimed that in the spacetime diagram, the area of the light rectangle is proportional to the

square of the proper time interval. I will demonstrate this by a geometrical argument on Monday. But given this

fact, it is easy to see that there is time dilation and that the twin effect occurs...

Tuesday May 4

- Radiation from point charges

- Lienard's generalization of Larmor's formula

- special case: linear acceleration; radiation pattern, "beaming"

- special case: circular motion: synchrotron radiation; example of charged particles in the Crab nebula

which emit synchrotron radition from electrons and positrons cycling in a magnetic field, with a spectrum

ranging all the way from radio frequencies up to 100 GeV, the latter being generated by particles with

kinetic energies of order 1500 TeV!

- radiation relaction: motivation/"derivation" for the Abraham-Lorentz force

Monday May 3

- Something I neglected to point out last Thursday: when we compute the retarded vector potential, the

leading order contribution to the radiation comes just from

(µ

Note that this integral is taken over the value of the current density at one time, t-r/c.

For a closed current loop this vanishes, but if div J is nonzero it does not. Griffiths invokes problem 5.7

to show that the integral is the time derivative of the dipole moment. A simpler way to see this is to write

the integral as a sum over charges times velocities:

∫ J = ∑ q

Dimensional analysis/physical picture of radiated power P = (µ

The dipole radiation comes from the dipole field "detaching", which happens at a distance of order the light

travel time in one cycle, l = c/f, which is also the wavelength, where f is the frequency. The power should scale as

(energy in the dipole at the distance l)/(period of a cycle). The energy density in the dipole goes as epsilon_0 times the

E^2, where E is the dipole field strength at the distance l, which goes as (1/epsilon_0)p/l^3. The total energy goes like

this density times l^3. Thus we have

P ~ [epsilon

Magnetic dipole and electric quadrupole radiation:

This comes from the next order term in the expansion from the distance dependence of the retarded time

in rho(r', t-|r-r'|/c) and J(r', t-|r-r'|/c). This arises from the first derivative term in the Taylor expansion of

the time dependence, and goes in the case of the current as (d/dt)J(r', t-r/c) rhat.r'/c. Compared to the

electric dipole contribution to the vector potential this is suppressed by the time derivative times r'/c, i.e. by

fd/c ~ d/l, where, as below, d is the size of the source. When computing the energy, the fields are squared,

so that the power in this radiation is suppressed relative to the electric dipole by (d/l)^2. The contribution

from rho at this order just goes into the electric quadrupole, but that from J sources both the electric

quadrupole and the magnetic dipole radiation.

- Showed this great demo and discussed its physics:

K8-42 RADIOWAVES - ENERGY AND DIPOLE PATTERN

Thursday April 29

- comment on fields of uniformly moving point charge: I pointed out what is happening in (10.69), where

the unit vector from the retarded position winds up relaced by the velocity vector divided by c.

- by the way, I remarked on what Griffiths calls the "extraordinary coincidence" that the field of a

uniformly moving point charge is directed from the present position of the charge rather than the retarded

position. This can be understand in a different way using the fact that in the rest frame

of the charge the field is just the Coulomb field, which is directed from the charge, then making

a Lorentz transformation on the fields. See Example 12.13 for a discussion of this. I wonder if there

is an even simpler argument along these lines...

- continuing with radiation: using these approximations (see Tuesday) in the retarded potential leads

to an expression containing the monopole potential, the retarded dipole potential, and the dipole radiation field,

etc, the last being from the time derivative of the dipole moment. The radiation field drops with one

inverse power of r rather than two, and dominates as long as lambda << r.

The electric and magnetic radiation fields must also fall of as 1/r, not 1/r^2.

When taking the derivatives of the potentials to obtain the fields, the 1/r parts

arise from the terms in which the derivatives act on the t or r in the argument of rho.

- No monopole radiation because of charge conservation. I made the comment that there are no dipole

gravitational waves because the time derivative of the mass dipole moment is the momentum, which is

also conserved. So the leading order gravitational waves are quadrupolar waves.

- Electric and magnetic dipole radiation fields: The electric field is proportional to minus (1/r) times

the projection of the second time derivative of the electric dipole moment orthogonal to rhat. The magnetic

field is perpendicular to both E and rhat, with magnitude E/c and direction such that E x B (and hence the Poynting

vector) is in the direction of rhat. We looked at angular dependence of the dipole radiation: zero along the

direction fo the dipole derivative, maximal perpendicular to that.

- The total power radiated is given by the simple formula P = (µ

derivative of the dipole moment, evaluated at the retarded time.

- For a point charge, pddot is the charge q times the acceleration, so we obtain the famous

Larmor formula P = (µ

- Decay of classical atomic orbits: it takes roughly a nanosecond. This is on the homework.

Tuesday April 27

- Point charges, Lienard-Wiechert potentials, electric and magnetic fields

- field of a uniformly moving point charge

- relativistic "pancake" field of rapidly moving charge

- radiation: the multipole expansion exploits the hierarchy of length scales often present in radiating

systems: d << lambda << r, where d is the source size, lambda is the wavelength, and r is the distance

from the source to the field point. Lambda is c/frequency. The first inequality thus says that the system

is must smaller than the distance light travels in one cycle. The key approximation following from

d << r is

|r - r'| = r - rhat.r' + O(d

and the key approximation from omega << c/d is

rho(r', t - |r - r'|/c) = rho(r', t - r/c) + rhodot(r', t - r/c) rhat.r'/c + O(rho (omega d/c)

Monday April 27

Lorenz gauge and retarded potentials: in Problem 10.8 Griffiths asks to confirm that the retarded potentials (10.19)

satisfy the Lorenz gauge condition which was assumed in deriving those potentials. I proposed a much simpler

way of doing this, but it's not quite conclusive. Namely, apply the d'Alembertian oeprator (box) to the Lorenz

gauge condition itself. To simplify notation I'll work with units in which mu_0 and epsilon_0 are unity. Then we have

box(div A + ∂

where the last equality follows from the cotinuity equation (local charge conservation). So the Lorenz gauge condition

expression div A + ∂

values and time derivatives of the function at a given time. So if the Lorenz gauge condition and its time derivative are

satisfied at one time, it will be satisfied for all time.

- Wire example

- Jefimenko's equations

Thursday April 22

Exam 2

Tuesday April 20

review for exam 2

Monday April 19

potentials in electrodynamics: writing B = curl A and E = - grad V - ∂

div B = 0 and curl E = - ∂

in terms of the potentials. The potentials are not uniquely determined: For any function

f of space and time, A + grad f and V - ∂

of the potentials is called a "gauge transformation".

The other Maxwell eqns involve the sources: charge and current densities.

They can be simplified by choosing the Lorentz (Lorenz) gauge condition, div A + (1/c

after which they take the simple form

box V = -rho/epsilon

where box = d'Alembertian = Laplacian - (1/c

If A' and V' don't satisfy the Lorentz gauge condition, then A' + grad f and V' - ∂

gauge transformation function f is chosen to satisfy the equation box f = s, where the source s

is s = - div A' + (1/c

all satisfy a similar equation.

The eqn box f = s has no unique solution, because we can add to f any solution of box f = 0.

This means that there is residual gauge freedom. That is, even after we impose the Lorentz

gauge conditions, there remains some freedom to change the potentials and still satisfy this

condition. As for the potentials, we need to supplement the equation by boundary conditions

to select a unique solution. The "retarded" boundary condition imposes that the potentials are

determined only by the values of the sources to the past. (The "advanced" boundary condition

is the opposite, with dependence only on the future.) We can write the general retarded solution:

see (10.19). I drew a spacetime diagram showing how this formula is interpreted: to find the

value of the field at a point in space and time, one integrates over the past light cone of that

point.

I showed that the integral (10.19) indeed satisfies the equation, by evaluating box [r

Using the Laplacian in spherical coordinates, and ignoring the trouble at the point r = 0,

this is zero (it is because s(t - r/c) satisfies the one dimensional wave equation (∂

If we do worry about r = 0 then we pick up a delta function from Laplacian r

So we have box [r

position r', and using that we can apply the box operator to the integral in (10.19). It comes inside

the integral, acting on the r and t dependence, and produces a delta function. The integral can then

be carried out, verifying that the equations (10.16) are satisfied.

Thursday April 15

- A bit more on RealD glasses: 1) If you place two pairs face to face, so the left eye of one pair

faces the right eye of the other, then the two quarter wave plates have their fast axes aligned,

so it becomes a half-wave plate, which has the effect of rotating the incoming horizontal

polarization to a vertical one. This is then blocked by the horizontal polarizer on the back side,

so the result is dark...but not completely, since there is chromatic dispersion. What you actually

see is a dark violet, and if you look at an angle you can see different color mixtures depending

on the angle. 2) If you place the two pairs face to face, and then rotate one by 90 degrees, it then

becomes the opposite quarter wave plate, and the two quarter wave plates cancel. They form

a sandwich between the now crossed linear polarizers on the backs of the glasses, so now the light

is blocked for all colors.

- Waveguides. I gave the highlights of Griffith's treatment, skipping some of the algebra.

K8-21 MICROWAVES - WAVE GUIDES |

K8-31 TRANSMISSION LINE SAMPLE |

A student asked about the limit of a large waveguide being equivalent to empty space, admitting

ordinary transverse plane waves. We can address that as follows. Consider the limit where the lengths

of the waveguide a and b are "very large". More specifically, consider a mode for which m/a and n/b are

very small compared to k, the wavenumber for the z-direction. Then the dispersion relation becomes

approximately omega^2 = c^2 k^2. But what about the z-components of the fields?

Refer to eqn (i) of (9.179), which comes from Faraday's law. This tells us that the magnitude of the

z-component of B scales as (m/a)/omega = (1/c)k_x/k_z times the transverse components of E.

It is thus suppressed by the ratio k_x/k_z compared to the transverse components of B in a plane wave.

A similar analysis works for the z-component of E, using (iv), which comes from the Maxwell-Ampere

equation.

One student pointed out something neat after class: The zig-zag bouncing waves model discussed in and

around Fig. 9.25 can also be exploited to "explain" why there is a z component of the magnetic or electric field,

and it even explains how large that component is: For example, if a transverse wave is travelling

"diagonally", with transverse B field (B_x,B_z), then similar triangles show that B_z/B_x = k_x/k_z.

This is just the relation inferred in the previous paragraph from Faraday's law.

I showed some great java applet simulations of waves in a waveguide, at falstad.com, which has loads

of other physics and math applets as well.

- The potential formulation of electrodynamics: I covered parts of section 10.1.

Tuesday April 13

- explained the notion of group velocity, and why it is equal to domega/dk.

- frequency dependence of the refractive index - covered the model in Griffiths

- how RealD 3D projection and glasses work. A digital signal with 24 frames per second

for each eye is tripled in frequency and alternated RLRLRL ... by a light projector, so each eye

sees the same image three times, each for 1/144 th of a second. The light from the projector goes

though a "Z-screen", which is an electro-optical liquid crystal modulator, that puts out the light

as right circular or left circular polarized, let's say for the right or left eye. This Z-screen must be

switchable at this high frequency, and synchronized with the incoming light sequence. Half the

light goes to each eye, and half of that is lost when it is circularly polarized, and the Z-screen

has some finite switching time, so less than 25% of the light makes it out of the projection

camera for each eye. The screen in the theatre is a "silver screen" that preserves the polarization

when it reflects the light, and it has a "gain" of brightness in the direction of the audience of

a factor of about 2.2-2.4 relative to a normal matte screen.

The glasses worn by the viewer pass one circular polarization through the right eyepiece, blocking the

opposite circular polarization. (Only 80% of the light of the correct polarization is transmitted, further

decreasing the intensity.) The left eyepiece passes the opposite circular polarization.

How do the eyepieces accomplish this? Each one consists of two layers: 1) a quarter wave plate

in front and 2) a horizontal polarizer in back. The quarter wave plate converts the incoming

circular polarizations into vertical and horizontal polarizations. The fast and slow axes for linear

polarizations of the quarter waveplates lie at 45 degrees, and are opposite for the two eyes,

so that opposite circular polarizations emerge with horizontal linear polarization.

I'm guessing the quarter wave plate is about 0.5 mm thick (but this could be totally wrong),

and the wavelength of the visible light is around 0.5 microns, so 1000 wavelengths fit.

In class I remarked that in order to produce the quarter cycle phase shift,

the fast and slow indices of refraction must differ by one part in 4000.

A student asked about chromatic dispersion, i.e. the fact that different colors would have

different indices of refraction, so the quarter wave plate wouldn't work. I answered that the

difference of indices would be of order 1%, which shouldn't present a problem. Well I think my

answer was confused! The wavelengths of different colors will differ by something of order unity,

so a quite different number of cycles will be completed for the different colors. Moreover,

If n_fast and n_slow differ from 1 by something of order unity, which probably they do, then

if their difference has to be tuned to one part in 4000 that tuning cannot possibly work for all

frequencies.

At the above link a design is mentioned that may be what is used, that seems to address part

of the tuning problem but not all: take two wave plates of the same

birefringent material but slightly different thicknesses L and L + ∆L, with their fast and slow axes

rotated 90 degrees. Then the second plate almost undoes what the first plate did, and the net phase shift

just comes from the difference in thickness. One polarization has a net phase change accumulation

k_slow L + k_fast (L + ∆L), and the other has k_fast L + k_slow (L + ∆L). These differ by π/2 if

(k_slow - k_fast)∆L = π/2. Now let's look at the frequency dependence. We have k = n(w)w/c,

where I use w in place of the angular frequency omega. So the condition reads

[n_slow(w) - n_fast(w)]w ∆L/c = π/2. Problem: we need a single value of the thickness difference

∆L that works for all frequencies w in the optical range, so we need [n_slow(w) - n_fast(w)]w to be

constant in this range! That's a significant demand since w changes by almost a factor of 2. So are there

materials whose indices of refraction behave this way? Maybe not. But the Wikipedia article also

mentions combining plates of different materials in order to cancel out the frequnecy dependence...

I don't know how it's actually done for these RealD glasses. If you find out please let me know!

Monday April 12

more on em waves in conductors

skin depth in silver at 6 E14 Hz (blue/green, wavelength 5000 A) is about 26 Angstroms,

whereas at 60 Hz it is about 8.5 mm.

skin depth in a wire: The electric field in an AC current carrying wire, and the current, are concentrated

on the surface of the wire at high frequencies. This is analogous to what we found for the plane wave

in a conductor. Another way to see it is to note that the oscillating B-field in the wire induces an E field

whose emf adds to the current on the outside of the wire and subtracts on the inside.

relation between skin depth and absorption? I did this in class but messed up with numerical factors.

but also with the energy dependence. The simplest way to compute the fraction absorbed is to compute the

fraction reflected, since there the ratio R is just the ratio of the squares of the electric field amplitudes.

Here is the correct argument:

According to (9.147) R = |(1-b)/(1+b)|^2, where (for typographical reasons) I write b for betatilde.

We are talking about the limit |b| >> 1, so we get approximately R = 1 - 4/|b| (get this by factoring

out a b from the numerator and denominator, then using the expansion (1 + 1/b)^n = 1 + n/b +... a

couple of times), hence T = 4/|b|. This transmitted part is absorbed in the conductor.

Now for vacuum/conductor interface, with µ approximately µ_0, we have from (9.146)

|b| = (c/omega)|ktilde| = |ktilde|/k_vac = (√2/2π)(lambda_vac/d_skin), so

T = (8π/√2) d_skin/lambda_vac. We learn that the ratio of skin depth to vacuum wavelength

controls the amount absorbed, but the numerical factor is pretty large, about 18.

With d_skin = 25A and lambda_vac = 5000 A we find T = 4π/√2 % = 9%. This is pretty close

to the 7% figure mentioned in the book.

An interesting question came up in class: how does the fraction absorbed depend on the

index of refraction of the transparent medium? For example, with a silver mirror covered with

glass, how does the index of refraction of the glass change the reflection coefficient?

The answer is that lambda_vac is replaced by lambda_glass, which is around 1.5 times smaller,

so there is 1.5 times as much absorbed, it seems... I think this can be understood as due to a better

impedance match. The wave speed in the conductor is very low. The speed in glass is lower than

in vacuum, so there is a closer match, hence more energy transferred.

Thursday April 8

oblique reflection and transmission:

Note that Fig. 9.14 and 9.15 are inconsistent with the fact (9.96, 9.97) that the components

of all three wavevectors in the plane of the interface are equal.

One point not made by Griffiths is that when µ

becomes equivalent to the statement that the reflected and refracted wavevectors are perpendicular.

In this configuration, the polarization vectors of these waves are also perpendicular, and are parallel

to the other wavevector. I have heard this given as an explanation of why there can be no reflected wave

at this angle of incidence, i.e. because there is no motion of the charges in the surface that can

produce both of these waves. This explanation sounds like it has some truth in it, but it can't

be really correct, because the true condition for no reflected wave is different if the magnetic

permeabilities of the two media are different. Also, one can ask why it it the reflected wave

rather than the transmitted one that is extinguished. Also, the transmitted wave is really a combination

of the incoming wave plus the wave generated by the charges in the material. Since the

incoming and transmitted waves are not parallel, the generated wave must not be parallel to

the transmitted wave. This appears to weaken the argument based on the polarization directions.

Perhaps there is a way to clarify and support this simple argument however.

em waves in conductors: I followed Griffiths, but here are a couple of notes:

The demonstration that the free charge density vanishes contains a subtlety. The argument goes like this:

the free charge density rho

∂

so rho

density does not decay exponentially! So what gives? The first two equalities are unassailable.

The problem lies in the last one: epsilon takes into account the "bound" charge, but what this

really means is the charge density arising from polarization. In a conductor, the ion charge is not

included in this polarization charge, so must be included in the "free" charge density on the

right hand side of the last term. Since it is constant in time, it can also be added to rho

time derivative term. So, what decays away exponentially is the sum of the ion and conduction

electron charges.

Another point not made explicitly in the book is that, for a good conductor, the imaginary

part of the square of the complex wavenumber (9.124) is much larger than the real part, so

on the complex plane ktilde^2 points almost along the imaginary axis. Therefore, its square

root, ktilde, lies at nearly a 45 deree angle, and has nearly equal real and imaginary parts, both

nearly sqrt(µ sigma omega/2)

Tuesday April 6

more on plane waves

em waves in matter

reflection and transmission

Monday April 5

electromagnetic wave equation & polarizations

K8-01
ELECTROMAGNETIC WAVE - MODEL |

M9-03 CIRCULAR POLARIZATION - STICK MODEL |

Thursday April 1

Perfectly conducting plasma and "Frozen-in theorem" a.k.a. "frozen field theorem" a.k.a. "Alfven's theorem" (see problem 7.59)

Implications: stellar collapse produces strong magnetic fields, folding/mixing fluid produces strong fields with rapid spatial variation

in direction. Magnetic reconnection.

Poynting's theorem, electic and magnetic field energy density, Poynting vector.

Tension of magnetic flux tubes: Consider a tube of flux lines with cross sectional area A and length L. If the plasma has approximately

constant density, then AL=const. as the tube is stretched. The frozen-in theorem says BA =constant, so B ~ L. The energy density in

the field is ~ B^2 _ L^2, and the volume is constant, so the field energy scales as L^2. This means that the tube acts like a spring

with a constant spring constant (I said this wrong in class). It takes energy to stretch it, and that energy is stored in the magnetic field.

It can be released when magnetic reconnection occurs.

Flow of energy into a resistor.

Momentum. rather than derive this as in the textbook, I appealed to the fact the momentum = energy/c for a massless particle.

(E^2 = p^2 c^2 + m^2 c^4, so if m=0, E = pc). This relation between momentum, energy, and speed, can be generalized:

Non-relativistically, momentum is mass times velocity, mv. Relativistically, the mass is replaced by (E/c^2), and the momentum

is p = (E/c^2)v. When v=c this becomes p = E/c. Thus the momentum density is the energy flux density divided by c^2,

S/c^2.

Examples of the charged hoop around a solenoid: the em field has circulating momentum, i.e. angular momentum. When the

electric or magnetic field is turned off, a torque is applied internally to the system and something starts rotating, so that

the angular momentum is conserved.

Tuesday March 30

How the Ruhmkorff coil works.

Discharge of coil: field strength needed to tear charges from surface is very high, probably

too high for the discharge to be initiated by that alone. Instead, I think that ions in the air

play a critical role...but I'm not positive (or negative).

Perfect conductors: in order not to have an infinite current the force must vanish, i.e. E + v x B = 0.

This means that the induced emf in a perfectly conducting circuit vanishes, so that the flux in the

circuit must be constant in time.

Example: square loop partially in a magnetic field (Problem 7.51).

Example: neutron star, treated as a conducting sphere spinning in a constant magnetic field along

the axis (Problem 7.45).

Maxwell's equations in matter. The current has a free part, a magnetization part that goes into

H, and a polarization part that goes into D.

Maxwell's conception of the displacement current: even the "vacuum" was a medium, the aether,

and the presence of an electric field entailed a displacement of some kind, such that when the

electric field is changing there is a changing displacement, hence a current.

(In quantum field theory, the vacuum, while Lorentz invariant, is a sort of medium, and can be

polarized, etc.)

Monday March 29

Energy stored in an inductor, 1/2 LI^2.

K2-21 RUHMKORFF INDUCTION COIL

Maxwell's displacement current term added to Ampere's law. First, in vacuum, then, in matter.

Thursday March 25

Example: H for a permanent magnet. Recall H = B/µ

I discussed the example of a cylindrical bar magnet with uniform magnetization M. Then div H = div M,

which is a delta function at the "caps" of the cylinder. So the H field of the magnet is identical to the

E field of a capacitor with two parallel circular plates. For a magnet much longer than it is wide, the

H inside is then much smaller* than M, so to a good approximation we have B = µ

B = µ

problem is to identify the bound current, and see what B that generates. The bound current is J

which is zero except for a surface contribution K

the cylinder like a solenoid, and the magnetic field is thus that of such a solenoid.

* Proof : The H from each cap is (1/4π) ∫ [(div M)/z

to the field point. At the midpoint of the cylinder, a distance z = h/2 from the cap, the contributions for the two

caps add to M(2 r

Magnetization and saturation: The permanent magnet on the lecture table is marked as having a field of

3.5 kilogauss, which is about 1/3 Tesla. A typical ferromagnetic material has a magnetic field that saturates

at a field of the order of a Tesla with 100% domain alignment. So it seems that magnet would be a few tens

of percent aligned. A small neodymium magnet by contrast has a field at the cap of order one Tesla.

Comments on Faraday's law curl E = - ∂

- The dimensions match. (The combination E + v x B in the Lorentz force law shows that the ratio

of the dimensions of E and B is velocity.)

- The parities match: both sides depend on the arbitrary right hand rule.

- The divergence of curl E is identically zero, which matches with the fact that div B = 0.

- In terms of the vector potential A, Faraday's law reads curl E = curl(-∂

of E is -∂

- The sign is essential for stability, otherwise there are runaway effects, whereby a changing flux induces more

of the same type of change, rather than opposing the change.

Lenz's law: the effects of electromagnetic induction always oppose the change that produced them.

Example: K2-62 CAN SMASHER - ELECTROMAGNETIC

The induced electric field generates a current in the can that is opposite to the current in the coil. The opposite

currents repel, and the can is pinched and repelled outwards. To understand the outwards force you have to

consider the flaring out of the magentic field of the coil: a field component perpendicular to the symmetry axis

is needed to produce the outward force along the axial direction.

Example: K2-44 EDDY CURRENT PENDULUM

Here a motional emf (in the rest frame of the magnet) generates currents in the pendulum. Once the current

begins to flow it feels a repulsive force opposing the change. When the penduum begins to exit the magnet

then it feels an attractive force, again opposing the change. In the (instantaneous) rest frame of the moving

pendulum, there is no velocity, but the magnetic field is changing, so there is an induced electric field, which

causes the current to flow.

Example: K2-61 THOMSON’S COIL

The primary coil has an AC current, and surrounds an iron core. The induced current in the secondary ring

opposes the changing flux. This thing can produce a steady nonzero average force, and therefore "floats" the ring.

I used to think this was simply explained just by analogy with the can smasher: when the flux increases, the induced

current opposes the change, hence is opposite to the primary current, hence repelled, and when the primary current reverses,

increasing the flux in the opposite direction again the induced current is opposite, hence still repels. This explanation

is bogus however, since it only covers two of the four quarters of a complete cycle. For instance in the second

quarter cycle, the flux is decreasing because the current in the primary is decreasing, so the induced current in the secondary

is in the same direction, opposing the decrease, and hence is attracted. Similarly it is attracted in the fourth quarter.

So something is amiss in this explanation. The missing thing is the contribution to the flux coming from the current in

the ring itself, i.e. the self-inductance! This opposes the change of secondary current. Without the self-inductance, the

current in the secondary lags that in the primary by 90 degrees in phase; with the self-inductance it lags by more, which

results in avoiding total zeroing of the average force. If the self-inductance dominates over the resistance, then the

secondary lags by 180 degrees, hence is always repelled. See Problem S7 of HW7 for details.

Tuesday March 23

Ferromagnetism - Started by "melting" the ferromagnetic domains of Nickel with this demo:

J7-13 CURIE POINT OF NICKEL

Some Curie point temperatures:

Nickel: 358 C

Cobalt: 1400 K

Iron: 1043 K (770 C)

Gadolinium: 16 C

Dysprosium: 85K (liquid nitrogen boils at 77 K)

Explained the nature of ferromagnetic domains and domain walls. The walls are several hundred

atoms thick. I can't find consistent statements about domain sizes. I read microns, but also millimeters

or fractions thereof. In any case, a domain is a part of the material where neighboring electron

spins are aligned, resulting in a mesoscopic magnetic moment. Only some materials have the

needed crystal structure. In those, the spin alignment occurs because that way the spin states are symmetric,

which implies that the spatial wave function is antisymmetric under electron intechange, which helps minimize

the electron-electron Coulomb electrostatic repulsion energy. If the energy is sufficiently lowered by this

arrangement, then the spins will beheld in alignment. This depends on temperature: if the termal energy

is too large, then the spin order will be melted. The transition where this happens is the Curie point.

Different domains form, oriented randomly with respect to each other, rather than one giant domain, because the

randomized domains have lower total energy. The cost is the domain walls where the spins don't get to

align perfectly. The balance of this effect with the overall magnetic energy determines the minimum total

energy domain size. In an external magnetic field, the domains aligned with the field grow at the expense

of other domains, and some domains flip, so the material gets a macroscopic magnetization. Some materials

can hold that magnetization for a long time, after then external field is removed. Those are called

"permanent" magnets. Lodestone, a natually occurring "permanent" magnetic rock, was presumably

magnetized by lightning strikes.

Ferromagnetic material when magnetized can provide a huge amplification of the original magnetic

field that triggered the magnetization. We demonstrated this with a superstrong electromagnet

fromed from a coil with a 1.5 V battery, and an iron core inside:

J6-04 LOW POWER - HIGH FORCE ELECTROMAGNET.

Einstein de Haas effect:

Quite remarkably, Einstein, together with de Haas (who was Lorentz' son in law whom Einstein had

helped get a temporary position), demonstrated experimentally in 1915 that ferromagnetism

is due to circulating currents that carry angular momentum. The method was to suspend an iron cylinder from a

fiber. Around the cylinder was a solenoid, in which an alternating current was driven, alternately magnetizing the

cylinder in opposite directions. The magnetization arises from circulating currents of charge...but also of mass.

So the magnetization also implies an angular momentum. Since there is no external torque on the cylinder, the

bulk material must aquire an equal and opposite angular momentum. By tuning the alternating current

frequency to the natural vibrational frequency of the torsional oscillations of the suspended mass, the resonance

can be used to enhance the effect. Einstein assumed the microscopic currents were due to cirulating electrons,

hence he used the gyromagnetic ratio obtained from the mass and charge of the electrons. They thought they

confirmed the predictions to within 10%, but that resulted from selective choice of data and massaging the

theoretical calculations. In fact, the prediction was off by a factor of 2, since it is electron spin, not orbital

motion that accounts for ferromagnetism, and the gyromagnetic ratio of electron spin is twice that for

orbital motion. Later experients saw this factor of two, even before electron spin and its associated

gyromagnetic ratio was discovered, but these experiments played no role in that discovery, which was

instead entirely based on atomic spectra.

Regarding this experiement, Einstein told a friend: "In my old age I am developing a passion for experimentation."

(He was 36 years old!) In fact, his physics, though fundamental, was always grounded in physical reasoning.

Perhaps his work at the patent office played a role in this. A case has been made for an important role for that

work in his discovery of relativity theory: Einstein's Clocks, Poincare's Maps, by Peter Galison. But an expert

historian of Einstein's work tells me that the case doesn't stand up to scrutiny. I think it's an interesting idea,

but the book is very repetitive.

Electrodynamics:

At first magnetism and electricity were disjoint phenomena. Then Oersted and Ampere showed that an electric

current (produced by a voltaic pile) produces magnetic effects. A natural question then was whether, conversely,

magnetic effects can produce an electric current. This was tried without success using static configurations,

but eventually Faraday found that a changing magnetic field, or moving a wire in a static magnetic field,

produces a current. I demonstrated this with K2-02 INDUCTION IN A SINGLE WIRE.

A useful quantity in characterizing the effects that produce a current in a circuit is the electromotive force,

or emf. I covered the definition and properties of emf as described by Griffiths.

A "motional emf" is produced by moving a wire in a time independent magnetic field. Quite generally

the motional emf is minus the rate of change of magnetic flux through the circiut loop. I proved this in the

way shown by Griffiths. If the loop is stationary but the magnet moves, the same effects result, since all

that matters is the relative motion. But if the wire does not move, it cannot be a magnetic force that makes

the current flow...so what can it be? It is an electric force...but unlike electrostatic forces, it can produce

an emf around a closed loop. So, it is a new kind of electric field, not derivable from the gradient of a potential,

whose integral ∫ E.dl around a loop bounding a surface S is equal to -(d/dt)∫ B.da, minus the rate of change of

flux through S. The loop integral of E can be written as a flux integral of curl E, and since this holds for

any surface S, we infer the differential form of Faraday's law,

curl E = - ∂

Monday March 22

Magnetic materials

Diamagnetism of liquid nitrogen (N2) and Paramagnetism of liquid oxygen (O2)

(link to a video demonstration). The difference comes down to electronic structure:

N has 7 electrons in the orbitals 1s (two), 2s (three), and 2p (three). The spins on the 1s

and 2s electrons are paired, but those on the 2p electrons are not. However, when two

nitrogen atoms bond to make N2, the six 2p electrons are shared, and the spins pair,

yeilding no net spin or orbital magnetic moment. In contrast, O has 8 electrons, i.e.

one more 2p electron. It turns out that the energy is minimized if the spins are NOT

all paired, so O2 has a net spin magnetic moment, hence is paramagnetic. The susceptibility

is mentioned in the table in Griffiths: 3.9 E-3 at -200C (= 73K, boiling point is 90K).

Diamagnetism of water is easily demonstrated using a dish of water, or with a

stream of water. (In the latter video, the strong attraction with the charged glass

rod is probably due to induced negative charge in the water stream, pulled from ground.

In any case, what I'm linking it for here is the last part, showing the repulsion from a magnet.)

A spectacular demonstration is levitating a water droplet, a strawberry, and even a frog,

in a 10 Tesla field. Pyrolytic carbon is similar to graphite but has a parallel planar structure

with macroscopically large planes. It has the largest diamagnetic susceptibility by weight

of any room temperature material, and can be levitated above small neodymium magnets.

Discussed the classical model of diamagnetism presented by Griffiths. Actually, it can

only be understood using quantum mechanics.

Ferromagnetism introduced by demo in which dysprosium (element number 66) is cooled to

liquid nitrogen temperature and undergoes a ferromagnetic phase transition, becoming strongly

attracted to a large permanent magnet:

J7-14 CURIE POINT OF DYSPROSIUM

Thursday March 11 Exam 1

Tuesday March 9 Review for exam.

Monday March 8

J5-17 HELMHOLTZ COILS AND HALL PROBE |

J7-11 PARAMAGNETISM AND DIAMAGNETISM |

Helmholtz coils: If the separation d = R then you get an especially uniform field near the

midpoint on the axis: the second derivative with respect to z vanishes. See problem 5.46.

A "Hall probe" is a magnetic field measuring device that uses the Hall effect to measure B, or rather

the component of B normal to the plane of the probe.

The earth's magnetic field is mostly produced by currents in the liquid outer core. The field outside the

surface of the earth is dominated by the dipole component, in fact it is 90% dipole by some measure...

You can read in detail about geomagnetism in these MIT Open Courseware lecture notes for Essentials of Geophysics.

The auxilliary field quantity H = (1/mu_0)B - M is introduced in analogy with the electric displacement.

Since curl M is the bound volume current density we have curl H = J

Jump conditions on H.

Demonstration of paramagnetism of copper sulfate (see above).

Linear, isotropic magnetic media: M = chi_m H. (N.B. we use the auxialliary quantity H here, unlike in

defining the dielectric susceptibility where we use P = chi_e epsilon_0 E.) The magnetic susceptibility

chi_m is a dimensionless number, positive for paramagnetic materials and negative for diamagnetic ones.

Thursday March 4

Ampere, after Oersted's and his own work showing that electric currents deflect a compass needle,

proposed that the magnetic field of the earth must arise from electric currents:

"...we could suppose that before we knew about the South-North orientation of a magnetic needle,

we already knew the needle's property of taking a perpendicular position to an electric current [...].

Then for one who tries to explain the South-North orientation, would it not be the simplest idea to

assume in the earth an electric current?"

Magnetic monopole moment vanishes (proved this), and dipole moment is 1/2 ∫ r x J dtau

(talked about what goes into proving this). For a planar loop showed the dipole moment is the

current times the area times a unit vector normal to the loop. For an arbitrary loop it is I ∫ da.

Proved that the integral of the area element is independent of the surface spanning the loop.

Jump condition for vector potential in Coulomb gauge.

Force grad(m.B) and torque m x B on a magnetic dipole. Illustrated with electromagnet coils

and permanent magnets. A dipole will be flipped by the torque until it aligns with the magnetic

field, then the force will pull it into the strong field region, where m.B is greater.

Looked at example of a pair of parallel current loops. The are physical (not pure) magnetic dipoles.

They attract. You can see this thinking about the parallel currents attracting. Or you can see it in

terms of the magnetic poles, or the magnetic dipole. As explained also on p. 257, I explained that

in a uniform magnetic field there would be no attraction, and it is only the flaring out of the field

sourced by one loop that can produce the net attractive force acting on the other current loop.

Magnetization: magnetic dipole moment per unit volume. Magnetized material has a curculating

bound surface current K

Tuesday March 2

Jump conditions for the magnetic field at a planar interface: div B = 0 implies the perpendicular component

of the magnetic field is continuous, and curl B = mu_0 J implies any jump parallel to the surface must be

perpendicular to the surface current K. Hence the jump of B must be perpendicular to both the surface

normal n and the surface current, i.e. it must be parallel to the cross product K x n. Setting up an example

shows that the coefficient is just mu_0: B^above - B^below = mu_0 K x n.

Vector potential, gauge freedom, Coulomb gauge.

In Coulomb gauge the vector potential generated by an element of current is parallel to that current.

Meaning of Coulomb gauge in the Fourier transform: div A = 0 implies k.a(k) = 0,

where lower case a is the Fourier transform of A, which is thus transverse.

Example of a spinning charged shell: constant magnetic field inside, pure dipole field outside.

The radius if the sphere is R and the angular velocity omega and surface charge density sigma.

Then the surface current density is K = sigma v phihat = sigma omega R sin(theta) phihat. Griffiths

(Example 5.11) gets the vector potential by integrating over the surface current. The magnetic field

is then obtained by taking the curl of the vector potential.

An alternate way to solve for magnetic field of the spinning charged shell: introduce a magnetic scalar potential

U, so that B = - grad U. This is possible as long as curl B is nonzero, so it can be done outside the sphere with

one function U^out, and inside with another U^in inside. Then div B = 0 implies that these each satisfy Laplace's equation,

and they are axisymmetric, so can be expanded in Legendre polynomials. Only positive powers of r occur in U^in,

only negative powers in U^out. The matching conditions come from the jump conditions for B, which imply (i) continuity

of the radial derivative, and a jump in the theta derivative proportional to the surface current. We can now virtually solve

this by inspection: the theta dependence of the surface current is sin(theta) = -(d/dtheta)P_1(theta). So only the P_1

term has a jump. This means that there can only be a P_1 term, since the absence of a jump in the r derivative and the

absence in the jump of the theta derivative are incompatible (check this). So we have

U^in = A(r/R) cos(theta), U^out = -(1/2)A(R/r)^2 cos(theta),

for some A, where the coefficient -1/2 was chosen so the r derivatives would match. U^in is proportional to z, hence

the interior magnetic field is constant in the z-direction. U^out has the form of a pure dipole potential.

To determine A we need to impose the condition that the negative of the gradient of U in the theta direction jumps

by mu_0 K = mu_0 sigma omega R sin(theta). This yields -1/2 A/R sin(theta) - A/R sin(theta) = mu_0 sigma omega R,

so A = -2/3 mu_0 sigma omega R^2.

Multipole expansion of the vector potential

Monday March 1

Force on a wire.

To what extent do the div and curl of a vector field determine the vector field?

One answer to this is the Helmholtz theorem, as described in Appendix B of Griffiths.

If div F = D and curl F = C, and if D and C fall to zero faster than 1/r

then there is a unique F that satisfies these equations and vanishes at infinity, which is given

explicitly by

F = -grad U + curl W,

where U(r) = (1/4π)∫ d

div F = 0 and curl F = 0, and F vanishes at infinity, then F vanishes everywhere.

A different approach is to assume the functions can all be Fourier transformed. Use the corresponding

lower case letters f, d, c, for the 3d Fourier transforms. (Since curl F = C, it must be that div C = 0, so k.c = 0.)

As I explained in class, div F = D and curl F = C are equivalent to ik.f = d and ik x f = c.

The first equation determines the "longitudinal" component of f and the second determines the "transverse" component.

We can write the solution for f:

f = (-i/k

This determines all but the k = 0 component, i.e. the constant part of F.

Solenoid with arbitary cross-section: Take the solenoid to run along the z-direction. If the transverse

components B_x and B_y are zero, then the loop integral form of Ampere's law immediately yields

B_z = mu_0 n I, where I is the current in the wire and n is the number of turns per unit length.

For the analogous toroidal problem, Griffiths uses the Biot-Savart law to show that the transverse

field is zero. But here is a nicer way to show it: split B into the components parallel and perpendicular

to the z-direction, B = B_z + B_perp, and note that by the symmetry neither of these depend on z.

Thus by itself div B_z = 0, hence since div B = 0, we also have div B_perp = 0. Also curl B_z has only

x & y components, and curl B_perp has only a z-component. The current has no z-component, so

curl B = mu_0 J implies curl B_perp = 0. We have shown that both the div and the curl of B_perp vanish

everywhere. As explained above, this implies that B_perp itself must vanish everywhere (except for a

possible constant part that is unrelated to the solenoid.)

A similar construction using cylindrical coordinates shows that the magnetic field of a toroidal solenoid is

purely in the angular direction, and is hence easily determined using the integral form of Ampere's law.

I suggest you verify this yourself.

Thursday Feb. 25

More on cycloid motion in crossed electric & magnetic fields. To understand the motion,

imagine the charge is moving perpendicular to the crossed fields at speed v_0 = E/B. Then the

electric and magnetic forces balance, so the particle is unaccelerated. In a proper relativistic

treatment, the speed must be less than c, the speed of light, so this undeflected motion is

possible only if E/B < c.

Next consider a more general motion, r(t) = v_0 t + s(t), which has acceleration equal to s'',

where the ' means time derivative. This should be equal to the force divided by the mass,

(q/m)(E + (v_0 + s') x B). We choose v_0 so that E + v_0 x B = 0, hence s'' = (q/m) s' x B.

So the motion described by s(t) is a cyclotron motion, which is added to the uniform velocity v_0.

Now suppose the particle is initially at rest. Then evidently s' starts out equal and oposite to v_0.

This motion is a cycloid.

Comment on magnetic forces do no work Example 5.3.

Oersted first showed that a magnetic compass needle is deflected by an electric current.

This showed that magnetism may be due to electric currents. A couple of months later

Ampere established the law of force between currents.

K1-01 FORCE BETWEEN CURRENT-CARRYING WIRES

The Biot-Savart law.

Currents: line, surface, and volume.

Continuity equation: div J + ∂

Magnetostatics: constant charge density and current density. Continuity equation then implies also div J = 0.

Griffiths derives div B = 0, and curl B = µ

equations, and I will skip that "derivation" (you are welcome to read it). Notice the second equation

makes sense only in magnetostatics, since div curl B is automatically zero, so we'd better have div J = 0.

Given enough symmetry, these equations can be used to find the magnetic field given the current.

Example 5.7: an infinite straight line current. Rather than appealing to the Bio-Savart law to get the

direction, cylindrical symmetry tells us all three cylindrical components of the magnetic field depend

only on the distance s from the axis. The curl equation then gives the phi component of B.

The curl eqn tells us there is no circulation of B around a rectangular loop in the z-s plane,

since no current flows through that loop. By symmetry the top and bottom edges cancel, so the

inner and our edges must cancel as well, so the z-component must be constant. A constant field

in the z direction can always be added but is not sourced by the line current. The

div equation tells us there is no magnetic flux through a cylinder centered on the axis. The top and

bottom caps cancel since the z-component of the field is constant (or vanishes), so the

radial component must vanish as well.

Tuesday Feb. 23

Started out with this demo:

J4-13 MATCHSTICK ON NICKLE UNDER GLASS

which illustrates how an electric field can polarize a neutral material

(in this case the wooden matchstick), and thus exert a force on it.

Then I charged a balloon by rubbing it with fur, after which it nicely stuck

to the door of the classroom, illustrating the same principle as above.

Examples of finding electric field given a polarization, or given a linear dielectric material

and an external electric field.

Jump conditions for displacement, constrasted with those for electric field.

Energy stored in system of linear dielectrics.

Magnetic (Lorentz) force on a moving charge, F = q v x B. This is orthogonal to the velocity,

hence does no work. It only redirects velocity, but does not add kinetic energy.

Example: cyclotron motion. Mentioned that these circular orbits are quantized in quantum physics,

and the corresponding energies are called Landau levels.

Example: Cycloid motion in crossed electric and magnetic fields.

Monday Feb. 22

A permanent dipole moment p in an electric field E has energy W = - p.E. But what if the dipole

moment is induced, say p = alpha E. Then we must integrate up from zero field to find the energy,

W = ∫ dW = ∫ -alpha E.dE = -1/2 alpha E^2. I used this in the discussion about computing alpha

from quantum perturbation theory in the 2/18 notes.

J4-22 PARALLEL PLATE CAPACITOR WITH DIELECTRIC

Discussed this demo. How to compute the field? I introdced here the electric displacement

D = epsilon_0 E + P. Approximate the dielectric using the linear, isotropic

relation between P = chi_e epsilon_0 E, in which case D = epsilon E, where

epsilon = epsilon_0(1+chi_e).

Terminology: chi_e is the electric susceptibility, while epsilon is the permittivity.

epsilon_r = epsilon/epsilon_0 = 1 + chi_e is the relative permittivity or dielectric constant.

Discussed uniformly polarized sphere.

Solved parallel plate capacitor with dielectric. Noted how capacitance is increased by a

factor of the dielectric constant.

Thursday Feb. 18

Examples of multipole moments, physical and "pure" dipoles, conducting sphere in an

external field produces a pure dipole field, electric field of a dipole.

Polarization (charge separation) of matter, polarizability, polarization density P

An "electret" is a material with a permanent electric polarization, analogous to a permanent

magnet. These can be fabricated by heating a material containing polar molecules in

an external electric field, then cooling the material until the molecular alignment

"freezes in". I read in Wikipedia that quartz can be in a naturally occurring electret

state, but I didn't find anywhere the magnitude of polarizations that can occur that way.

The textbook mentions in Problem 4.11 that barium titanate is the "most familiar"

electret that holds its own polarization (as a reset of its intrinsic structure). If quartz does

as well, this is puzzling, since quartz seems more familiar! Note that an electret would

tend to neutralize its field, by accumulating opposite charges from the environment on its ends.

The dipole moment of a conducting sphere in an external electric field can be read off

from Example 3.8. Dividing by the field strength yields the polarizability, which is

R^3(4π epsilon_0). It's interesting to compare this to the table 4.1 of atomic polarizabilities.

Here I'm getting a bit carried away...

Dividing through by 4π epsilon_0, they are, in units of cubic Angstroms (10

0.667 for H, 0.205 for He, and 24.3 for Li.

I was wondering how to account for the differences.

One can calculate the polarizability using quantum mechanics. The energy of a permanent

dipole moment p aligned in an electric field of strength E is -pE. If the dipole moment is

induced by the electric field, p = alpha E, then the energy is the integral of -alpha E dE,

i.e. it is -1/2 alpha E^2. So the polarizability is minus twice the energy shift divided by the squared

field strength. For the ground states of the atoms in the table, I think the first order shift

vanishes by symmetry, so the leading order comes from second order perturbation theory.

The result for H is 9/2 (a_B)^2, where a_B = 0.53 A(ngstroms) is the Bohr radius, which

yields 0.670 A^3. This differs from 0.667 by only 3/667 0r about half a percent. So for this

simple atom, lowest (second) order perturbation theory gives an excellent approximation.

For He, despite having two electrons, the result is less than a third as large.

I mumbled in class something about the Pauli exclusion principle, but this is nonsense,

since in any case the electron spins are different so there is no exclusion of their spatial states.

I think the real issue is that the jump to the lowest excited state in Helium is roughly twice

as large as in H, because of the greater nuclear charge (although once the electron jumps

outside the ground state orbital the other electron screens the nuclear charge). In second order

perturbation theory this energy jump is squared, so this presumably accounts for the different

polarizability.

For Li, on the other hand, as pointed out by a student in class, there are 2p spatial states available

that lie rather nearby in energy. These lie only about 1.5 eV above the ground state, compared

with the jump of about 10 eV for H. (They would be degenerate were it not for the screening effect

of the two 1s electrons.) The contribution from these is thus (10/1.5)^2 = 44 times larger than

for H. Applying this factor to the polarizability of H (0.667) we get about 30. The actual

polarizability of Li is 24.3. The fact that it is not as high as 30 makes sense, because the energy

shift in second order perturbation theory comes from summing over all the states, and as we

go to the higher levels (3s,p,d, etc) the energy jumps approach those of H, since the two

1s electrons screen the nuclear charge.

Tuesday Feb. 16

Legendre polynomials, multipole expansion

Note that a single charge at the origin has only a monopole moment, but the same single charge

displaced to another point has nonzero multipole moments. The potential it generates is proportional

to 1/|r - r'|, which is a solution to Laplace's equation, hence can surely be expanded in the general form of (3.65),

∑(A

To find the B

1/|r - r'| = 1/(r-r') = (1/r)∑(r'/r)

so we read off that in that case, B

i.e. we replace theta by theta', where theta' is the angle between r and r'. Thus

1/|r - r'| = (1/r) ∑(r'/r)

In deriving the general form of the multipole expansion, (3.95), Griffiths expands 1/|r - r'|,

finding "the remarkable result" that it is equal to the expression above. But since

he already derived the general form (3.56), it shouldn't seem so "remarkable".

Monday Feb. 15

Separation of variables, cartesian and spherical

Monday Feb. 8, Tuesday Feb. 9, Thursday Feb. 11: SNOWED OUT

Thursday Feb. 4

DEMOS:

J2-11 FRANKLIN’S WHEEL |

J3-22 FARADAY CAGE - ELECTROSCOPE |

Charge distribution on a conducting disk or needle: I found an extremely nice, two page paper on this:

R. H. Good, American Journal of Physics 65, 155 (1997). He shows that the charge distribution which

produces a zero field on the disk is obtained by projecting a uniform charge distribution on a spherical shell

onto the equatorial plane. This yields a non-uniform charge density on the disk, divergent at the edge.

Similarly the distribution on a needle is obtained by projecting the spherical shell onto an axis, and this

yields a uniform charge density.... which seems paradoxical, since it seems as if the field at any point

except the midpoint could not vanish. See the paper for the explanation!

Method of images (cf. section 3.2 Griffiths) - Covered as in Griffiths. Regarding the plane, he says the induced charge

is -q, as "you can perhaps convince yourself it had to be." The best argument for this that I could find so far is to consider

a Gaussian surface that is a large disk under the conducting plane, then follows the field lines around behind the charge

q, and then jumps across. In the limit that this surface goes to infinity in all directions, the flux will vanish, so the total induced

charge must cancel the charge q. For a picture of the field lines for the plane and sphere problems see

http://www.scielo.br/img/revistas/rbef/v31n3/916fig01.gif

Tuesday Feb. 2

- Computational note: to find the leading order behavior of a function of z, when z >> R,

express the function in terms of the small dimensionless ratio R/z, and make a Taylor expansion about R/z = 0.

- Conductors (cf. section 2.5 Griffiths). Note that charges are bound to a conductor by some potential well,

so charges are confined on the conductor unless the electric field is really large. Surface charge density

is determined by the jump condition (since the field inside vanishes) to be epsilon_0 times the outward

normal component of the electric field at the surface (there is no tangential component).

[Here is the talk I mentioned in class by David Griffiths on the charge distribution on a conductor in

different dimensions: http://online.itp.ucsb.edu/online/utheory03/griffiths/.]

- Uniqueness theorem for solutions to Poisson's equation:

Suppose div grad V

Then the difference W = V

Then div(W grad W) = (grad W).(grad W). Now integrate both sides of this equality over R.

Using the divergence theorem, the left hand side becomes the flux of W grad W through the

boundary of R. If the boundary conditions imply that this vanishes, then it follows that grad W =0,

i.e. W is constant. We consider three versions of this theorem:

1) Suppose V

Then W = 0 on the boundary. So W = constant in the region, and since W = 0 on the boundary, it must

actually be zero everywhere in R. Thus V

2) Suppose the boundary condition is that grad V

(Neumann boundary condition). Then the boundary integral in the above equation vanishes, so again

W = constant, i.e. any two solutions (if they exist) differ at most by a constant.

3) Suppose the boundary is the surface of a conductor or multiple conductors, so that V

are each constant on all components of the boundary, and by Gauss' law the flux of grad V

over any component of the boundary are equal to each other and equal to -Q/epsilon_0, where Q is the total

charge on that conductor. Then the flux of (V

so again we conclude that W = constant, so the potentials differ at most by a constant.

- Solutions to Laplace's eqn have no local minima or maxima: If div grad V = 0 then the flux of grad V through

any closed surface is zero. Suppose V has a local minimum at a point p. Then grad V points away from

p in all directions, so the flux of V through a small sphere surrounding p is positive. This contradicts the

fact that it must be zero. Hence there cannot be a local minimum. Similarly there cannot be a local maximum.

This gives an alternate proof of the fact that if a solution V to Laplace's equation is constant on the boundary

of a compact region R, then it is constant everywhere throughout R, otherwise it would have a local maximum

or minimum inside.

- Examples: Applied the uniqueness theorem(s) to the examples discussed in section 2.5.2. As pointed out to me after

class, I overcomplicated the argument for example 2.9. The potential is constant on the conductor, hence

constant on the spherical outer surface. There is a spherical solution to Laplace's equation outside the conductor

satisfying this boundary condition, and by the uniqueness theorem this must be the solution. Thus the charge density

at the surface is spherically symmetric as well, i.e. the charge q is spread uniformly over the sphere, and contributes

nothing to canceling off the field in the interior of the conductor.

- Capacitors: given an arrangement of two conductors, carrying charge Q and - Q, the potential difference ∆V

between the conductors must be proportional to Q. Proof using the uniqueness theorem: given a charge distribution

and potential satisfying Laplace's equation and conductor boundary condition for one value of Q, a solution is

obtained for 2Q simply by scaling the charge density and the potential everywhere by a factor of 2. By the uniqueness

theorem, this must be the actual solution. Thus Q = C ∆V for some constant C, the capacitance, determined by the

conductor geometry. The SI unit of capacitance is the farad (F), 1 F = 1 C/V. A single conductor is assigned a capacitance

using for ∆V the potential difference between the conductor and infinity.

Monday Feb. 1

- Finding V from E: the potential change from a to b is minus the line integral of E along any curve from a to b.

For example, taking V = 0 at infinity, V(r) is minus the line integral of the electric field from infinity to r.

- Work and energy in electrostatics - cf section 2.4 of Griffiths.

Thursday Jan. 28

Applications of Gauss' law: exploit symmetry. We did this for a spherically symmetric

charge distribution and showed that the field outside is the same as if all the charge were

concentrated at the center. We also showed that the field inside a spherical shell is zero.

The field of an infinite plane is constant, independent of the distance from the plane.

It's magnitude is sigma/2epsilon_0, and direction is away from the plane if sigma > 0.

Jump Conditions: The field flips direction from one side of an infinite plane to the other,

so the field itself jumps by sigma/epsilon_0 in the direction perpendicular to the plane.

This jump condition holds across any surface charge. For example, across a spherical

shell, the field jumps from 0 inside to sigma/epsilon_0 in the radial direction outside.

Spherical shell versus infinite plane: We got into an interesting discussion in class,

sparked by my erroneous statement that the field very close to any surface charge is

the same as the field very close to an infinite plane, i.e. sigma/2epsilon_0. The example

of a spherical shell, which has a field twice as large as this just outside, shows this to be wrong.

One student suggested that this discrepancy could be understood by thinking of a contribution

sigma/2epsilon_0 coming from the nearby cap of the sphere, and an equal contribution

coming from the rest of the sphere. This is right in the following sense. Consider a point

at a distance d from the north pole of a sphere of radius R, and consider a cap of radius s.

If you take the limit in which both d and s tend to zero, and d goes to zero faster than s,

i.e. d/s also tends to zero, then the contribution from the cap indeed limits to the planar result

sigma/2epsilon_0.

Electrostatic potential: A Coulomb field r/r

a function, -grad(1/r). A superposition of Coulomb fields is thus minus the gradient of

a sum of 1/r functions centered at the different charges. That is, E = -grad V for some

function V, called the electrostatic potential. The SI unit of potential is the volt (V):

1 V = 1 Nm/C = 1 J/C.

I was asked in class about the expression for the electric field as an integral over Coulomb

fields for a continuous charge distribution. This has a script r squared in the denominator,

which blows up when r = r'. I said this does not lead to an infinite result, because it is multiplied

by the script r unit vector, which points in all directions near r = r' and hence the contributions cancel.

While this is true, there is another reason why no infinity occurs: the volume element in the

neighborhood of that point also goes as script r squared, which cancels the former one.

One could ask a similar question about the scalar potential. Suppose we are integrating

the function 1/r over a volume including the origin r = 0. Do we get infinity? No: the volume

element in spherical coordinates is r

volume element goes as r, not 1/r.

Tuesday Jan. 26

Superposition principle: the electric fields from multiple charges add vectorially.

This simplifies things a lot! It is not exactly true: due to the dynamical

nature of the vacuum in quantum field theory there are corrections, tiny under

normal circumstances. But in strong enough fields these corrections can become

important, even dominant. An example is the scattering of light by light:

two photons each split into electron-positron pairs, which then annihilate with

each other pairwise to produce two new photons. This becomes important

when the energy of the field in a volume of the electron Compton wavelength

cubed, times the square of the fine structure constant, is comparable to the

rest mass of an electron times the squared speed of light. This turns out to be

an electric field of about E18 V/m, or a magnetic field of about E9 T.

Magnetic fields of order E10 T are believed to exist around certain neutron

stars, called magnetars.

Gauss' law: First we considered the flux of the electric field through a sphere

centered on a point charge q, and found it is equal to q/epsilon_0, independent

of the radius of the sphere. Then we argued it is also independent of the shape of

the surface. Therefore the flux through any surface is the total charge enclosed,

divided by epsilon_0. Using the divergence theorem, and the fact that this is true

for any surface, we infer that div E = rho/epsilon_0, where rho is the charge density.

We should be able to derive Gauss' law instead by just computing the divergence

of the electric field of a charge distribution, given the integral expression for that.

This involved computing div(r/r

But this can't be right at r = 0, since integrating this divergence over an arbitrarily

small sphere around r = 0 yields, with the help of the divergence theorem, 4π.

The resolution: div(r/r

Monday Jan. 25

Course prerequisites: Phys 171, 272, 273, 374; Math 140, 141, 240, 241, 246 (or 414)

Supplementary books on vector calculus:

Div, Grad, Curl and all That: An Informal Text on Vector Calculus, H.M. Schey

A Student's Guide to Maxwell's Equations, Daniel Fleisch

I will essentially skip Chapter 1. Please review the math and make sure you know it,

and learn it if you don't know it. I am happy to answer any questions you have about it.

Electromagnetism is now understood as part of a (partially) unified electro-weak interaction,

based on a "gauge theory" of the group SU(2)xU(1). This symmetry is broken by the Higgs

field, and except for the electromagnetic force all the others become short ranged. We have

a subject of classical electromagnetism because the force is long ranged, and macroscopic fields

due to multiple sources add together coherently.