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CONTENTS


"UBC Physics 108 [Spring 2004]" Course

"DC Circuits" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"Current & Resistance"

by Jess H. Brewer on 2003-02-09:

Drude Theory

Ski Slope Analogy: Idealized skis: frictionless. Idealized skiers: indestructible morons. Idealized collisions: instantaneous, perfectly inelastic.

Drift velocity vd = acceleration (eE/m) mean time between scattering collisions .

Problem: how can be independent of the "steepness of the slope"? Short excursion into Quantum Mechanics and Fermi-Dirac statistics: electrons are fermions (half-integer spin) so no two electrons can be in the same state. Thus the lowest energy states are all full and the last electrons to go into a metal occupy states with huge kinetic energies (for an electron) comparable to 1 eV or 10,000 K. Only the electrons at this "Fermi level" can change their states, so only they count in conduction. So our ideal skiers actually have rocket-propelled skis and are randomly slamming into trees (and each other) at orbital velocities (we will neglect the problems of air friction); the tiny accumulated drift downhill is imperceptible but it accounts for all conduction.

J = "flux" of charge = current per unit perpendicular area (show a "slab" of drifting charge) so J = n e vd, where n is number of charge carriers per unit volume. (For Cu, n is about 1029 m-3.)

Ohm's Law

J = E, defining the conductivity = n e2 /m, measured in Siemens per metre (S/m) if you like SI units (1 S = 1 A/V). I don't.

For Cu, is around 108 S/m. Putting this together with n, e and me = 9 x 10-31 kg, we get ~ 10-13 s. At vF ~ 106 m/s this implies a mean free path ~ 10-7 m. Compare lattice spacing ~ 10-10 m. The drift velocity vd is only ~ 10-3 m/s.

In semiconductors, n is a factor of 10-7 smaller and vd is a factor of 107 larger, almost as big as vF! So in some very pure semiconductors transport is almost "ballistic", especially when the size of the device is less than .

Briefly discuss superconductors.

The inverse of the conductivity is the resistivity , measured in Ohm-meters. (1 Ohm = 1 V/A).

Use a cartoon of a cylindrical resistor of length L and cross-sectional area A to explain how this works, giving R = L/A and the familiar V = I R.


"RC Circuits"

by Jess H. Brewer on 2003-02-11:

RC is a Time Constant

Consider that R is measured in Ohms = Volts/Amp = Volt-sec/Coul whereas C is measured in Coul/Volt; thus RC is measured in seconds. This simple dimensional analysis should make you suspect that there is a time constant that grows with both C (charge capacity) and R (resistance to the flow of charge).

Discharging a Capacitor Through a Resistor

Start with an open circuit with a charged capacitor connected to a resistor; now close the switch. What happens? Well, basically, the charge is going to bleed off the capacitor through the resistor. The bigger the capacitor, the more charge there is to bleed off (for a given initial voltage drop across the capacitor) and the bigger the resistor, the slower it bleeds. The voltage on the capacitor is propostional to its remaining charge and the voltage drop across the resistance (proportional to the current, which is the rate of change of the charge on the capacitor) is exactly balanced by the voltage on the capacitor (because potential is single-valued, see Kirchhoff's Rule #2). So you pretty much know just what to expect; it is just like a projectile slowing down under viscous flow: exponential decay with a mean lifetime = RC.

Now do it mathematically: solve the differential equation for Q(t).

Charging Up a Capacitor Through a Resistor with a Battery

Start with the capacitor uncharged and then apply a constant voltage through a resistor. What happens?


"RC with AC"

by Jess H. Brewer on 2003-02-14:

. . . or we might have a Preview of AC circuits:

Driving an RC Circuit with an AC Voltage

Imaginary Exponents



"Diffraction" Topic

Found 4 Lectures on Mon 07 Oct 2024.

"Waves Bend Around Corners!"

by Jess H. Brewer on 2003-03-25:

Huygens' Principle

"All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. At a later time, the new position of the wavefront will be the surface of tangency to these secondary wavelets."
In other words, waves bend around corners!

Less obvious is the fact that a wave also interferes with itself even if there is a continuous distribution of sources.

Diffraction Pattern from a Slit of Finite Width

To see this in the simplest case, take waves coming through a single slit of width a and divide up a into N equal "pseudoslits" a distance d = a/N apart. We can then use the formula above and let N to get (after several tricky steps) the result

I = I0 sin2 /2

where

= (a sin /).

Some features of this result:
  1. At the central maximum ( = = 0) One sees the full I0. This can be seen from l'Hospital's rule on (sin x)/x as x goes to zero.

  2. The intensity goes to zero at any nonzero for which sin = 0. This occurs for any integer multiple of . The first minimum of the diffraction pattern occurs when = , which in turn implies

    a sin 1 = .

  3. The secondary maxima of the diffraction pattern can be found by setting the derivative of I with respect to equal to zero (condition for an extremum). The resultant formula contains both a term that goes to zero at the minima (zeroes) and another term that reduces to

    = tan .

    This transcendental equation can easily be solved by plotting both x and sin x on the same graph and looking for intersections. Don't go looking for an analytical solution.

That's enough for one day.


"Waves Bend Around Corners!"

by Jess H. Brewer on 2003-04-02:

Huygens' Principle

"All points on a wavefront can be considered as point sources for the production of spherical secondary wavelets. At a later time, the new position of the wavefront will be the surface of tangency to these secondary wavelets."
In other words, waves bend around corners!

Less obvious is the fact that a wave also interferes with itself even if there is a continuous distribution of sources.

Diffraction Pattern from a Slit of Finite Width

To see this in the simplest case, take waves coming through a single slit of width a and divide up a into N equal "pseudoslits" a distance d = a/N apart. We can then use the formula above and let N to get (after several tricky steps) the result

I = I0 sin2 /2

where

= (a sin /).

Some features of this result:
  1. At the central maximum ( = = 0) One sees the full I0. This can be seen from l'Hospital's rule on (sin x)/x as x goes to zero.

  2. The intensity goes to zero at any nonzero for which sin = 0. This occurs for any integer multiple of . The first minimum of the diffraction pattern occurs when = , which in turn implies

    a sin 1 = .

  3. The secondary maxima of the diffraction pattern can be found by setting the derivative of I with respect to equal to zero (condition for an extremum). The resultant formula contains both a term that goes to zero at the minima (zeroes) and another term that reduces to

    = tan .

    This transcendental equation can easily be solved by plotting both x and sin x on the same graph and looking for intersections. Don't go looking for an analytical solution.

That's enough for one day.


"Resolution"

by Jess H. Brewer on 2003-04-04:

Circular Aperture

Think of it as a "square slit with the corners lopped off". This makes it "effectively narrower" which makes the diffraction pattern wider. The numerical result (to be memorized, sorry!) is

a sin 1 = 1.22 .

Babinet's Principle

An obstacle is as good as a hole.

Picture light shining through a large aperture and consider the region "straight ahead" of the aperture (i.e. neglect the fuzzy areas around the edges of the region in shadow).

If you place a small obstacle in the middle of the aperture, you are subtracting the amplitude contributions of the rays that would have arrived at the final screen from where the obstacle is now.

Now take away the obstacle and instead block off the whole aperture except for a hole of the same shape as the former obstacle (and in the same place). Now only the rays that were formerly being blocked are allowed through.

Since amplitudes are squared to get the intensity, a "negative" amplitude is just as good as a "positive" one. Thus the two situations give the same diffraction pattern on the final screen. There will be a bright spot directly behind the obstacle, just as you would expect for the hole.

Detector Arrays

Interference is reversible. Just run the rays backward. Thus a telescope (for instance) "sees" diffractive rings around a distant star; the width "fuzziness" of the star (the size of the dot it makes on the final optical detector of the telescope) is larger for a smaller telescope diameter. A telescope whose resolution is limited by this effect is called "diffraction limited" and is considered a pretty good telescope if it is a big one.

The width of a diffraction pattern is defined as the angular distance from the central maximum to the first minimum.

Rayleigh's Criterion

Two objects can be resolved if the central maximum of one falls on the first minimum of the other.


"Dispersion"

by Jess H. Brewer on 2003-04-07:

Dispersion

The wavelength (colour) dependence of the interference pattern from a grating determines how useful it will be for resolving sharp "lines" (light of specific wavelengths) in a mixed spectrum. The rate of change of the angle of the mth principal maximum with respect to the wavelength is called the dispersion Dm of the grating. This is easily shown to have the value

Dm = dm/d = m/d cos m .

Note that there is a different dispersion for each principal maximum. Which m values will give bigger dispersions? Why does this "improvement" eventually have diminishing returns?

Resolving Power

A separate question is How close together () can two colours be and still be resolved by the grating?

Well, the two lines will just be resolved when the mth order principal maximum of one falls on top of the first minimum beyond the mth order principal maximum of the other.

By requiring the path length difference between adjacent slits to differ (for the two colours) by /N (where N is the number of slits) we ensure that the phasor diagram for the second colour will just close (giving a minimum) when that of the first colour is a principal maximum. This gives a resolving power

Rm = / = m N .



"Electrostatics" Topic

Found 6 Lectures on Mon 07 Oct 2024.

"The Electric Field"

by Jess H. Brewer on 2003-01-23:

Electric Dipoles

Show attempted 3D visualization. Calculate torque in terms of dipole moment p = q d.

Line of Charge

Calculate field a distance r away from [the centre of? VOTE!] a finite line charge of length L. Treat limiting cases r >> L and r << L.


"Gauss' Law I"

by Jess H. Brewer on 2003-01-26:

What is Coulomb's Law "saying" about the flux of "electric field lines"? (Work backwards to Gauss' Law.)

Extend to more complex isotropic charge distributions.

.

.

.


"Gauss' Law II"

by Jess H. Brewer on 2003-01-27:

Conductors

Gauss' Law with all the Constants


"Potential"

by Jess H. Brewer on 2003-01-30:

Electrostatic Potential

Notation: I will use V here instead of ["phi"] (chosen in class) because HTML still has no Greek letters except "µ".

In principle, it's easier to find E from V than vice versa, because it's a lot easier to integrate up a scalar function than a vector one! (And derivatives are easy, right?) However, in practice (at the level of P108) we are not going to be evaluating arbitrary, asymmetric charge distributions, but only the simple symmetric shapes and combinations thereof (using the principle of additive superposition). In these cases Gauss' Law allows us to find E easily and find V by simple integrations; so that's mostly what we do.

Examples


"Hammers, Doors, Capacitance & Dielectrics"

by Jess H. Brewer on 2003-02-01:

Hammers & Doors: Vector Calculus

Strictly optional (alternate, more compact notation).

When you first encountered Algebra it gave you new powers -- now you could calculate stuff that was "magic" before. (Recall Clarke's Law.) This is what I call The Hammer of Math -- "When all you have is a hammer, everything looks like a nail."

This year (or maybe earlier) you have discovered that algebra is also a Door -- the door to another whole world of Math: the world of Calculus. Now you are exploring a new, different Hammer of Math -- the hammer of calculus, with which you can drive a whole new class of nails!

This cycle never ends, unless you give up and quit. Every year you will have a new Door of Math opened by the Hammer your mastered the year before. Next year you will probably go through the Door of Vector Calculus to find elegant and powerful Hammers for the nails of vector fields. I am not supposed to tell you about this, because it's supposed to be too hard. So I won't hold you responsible for this topic, but I gave you a handout on it (and will discuss it a little in class) because you deserve a glimpse of the road ahead. Think of it as a travel brochure that shows only the nice beaches and night clubs.

Topo maps and equipotentials: meaning of the gradient operator.

Capacitors and Capacitance

(Textbook's Ch. 30) Capacitance C is a measure of a capacitor's capacity to hold charge (for a given voltage between the plates).

Thus Q = C V or V = (1/C) Q.

Units: a Farad (F) is one Coulomb per Volt.

Start with simplest (and most common) example, the parallel plate capacitor: this case defines the terms of reference clearly and is in fact a good approximation to most actual capacitors. Know the formula by heart and be able to derive it yourself from first principles! The capacitance of a parallel plate capacitor of area A with the plates separated by d is given by Cpp = A/d.

The capacitance of a capacitor consisting of concentric spherical shells of radii a and b is given by Csph = 4 [(1/a) - (1/b)]-1.

Capacitance of the Earth: treat the Earth as a conducting sphere of radius RE = 6.37 x 106 m. If the "other plate" is a concentric spherical conducting shell at infinite radius, what will be the potential difference between the "plates" when a charge of Q is moved from the shell at infinite radius to the Earth's surface? Answer: 710 µF (later on I will show you a capacitor you can hold in the palm of your hand that has a thousand times the capacitace of the Earth!) Note: this is not the same thing as you calculated in the 3rd homework assignment.

Pass around a 1 F capacitor -- more than 1000 times as big as the Earth!

The capacitance of a capacitor consisting of concentric cylindrical shells of radii a and b and equal length L is given by Ccyl = 2 L/ln(b/a).

Note that in each case C = (numerical constant) (distance). Check that this makes dimensional sense.


"Electrostatic Springs and Energy Storage"

by Jess H. Brewer on 2003-02-04:

Capacitor as an Electrostatic "Spring"

If you like you can think of 1/C as a sort of "electrical spring constant": if you move Q away from its equilibrium value (zero) you get a "linear restoring voltage".

Arrays of Capacitors

An arbitrary network of capacitors can always be replaced by a single equivalent capacitor.

An array of capacitors in parallel has an equivalent capacitance equal to the sum of their separate capacitances. [Explain.]

An array of capacitors in series has an equivalent inverse capacitance equal to the sum of their separate inverse capacitances. [Explain.]

Dielectric Materialism (Ch. 29)

Basically just replace by = (where is the dielectric constant, a pure number always 1) and everything takes care of itself. Thus C always gets bigger (by a factor of ) when there is a dielectric in between the plates. [Explain.]

Electrostatic Energy Storage

Recall the question at the beginning: why isn't a big capacitor a good replacement for a battery? Because the voltage decreases with the remaining charge! This has other implications as well....

The energy required to put a charge Q on a capacitor C is not just VQ! The first bit of charge goes on at zero voltage (no work) and the voltage (work per unit charge added) increases linearly with Q as the charge piles up: V = (1/C) Q. Thus dU = (1/C) Q dQ. Integrating yields U = (1/2C) Q2 or U = (1/2)C V2.

For a parallel plate capacitor, V = E d and C = A / d. Thus U = (1/2) AE2 d. But A d is the volume of the interior of the capacitor (the only place where the electric field is nonzero). Thus if u is defined to be the energy density per unit volume, then we have u = (1/2) E2. "It turns out" that this prescription is completely general! Wherever there is an electric field, energy is stored at a density u given by the formula above.

It is now getting really tempting to think of E as something "real", not just a mathematical abstraction.



"Elementary Particles" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Elementary Particles"

by Jess H. Brewer on 2003-04-08:



"EXAM" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"First Midterm"

by Jess H. Brewer on 2003-02-06:


"Second Midterm"

by Jess H. Brewer on 2003-03-13:



"Faraday & Inductance" Topic

Found 4 Lectures on Mon 07 Oct 2024.

"Magic!"

by Jess H. Brewer on 2003-03-05:

"Deriving" Faraday's Law:

Consider a metal bar of length L moving sideways through a uniform perpendicular magnetic field B: assuming positive charges can move, use Hall effect to find resultant voltage between ends of the bar: VHall = v B L where v = dx/dt is the speed of the bar. Now close the loop with a wire that goes outside the field region and let current flow. The direction of the current will be such as to make its own field either parallel or antiparallel to the original B depending on whether the loop is moving into or out of the field region. (This is the essence of Lenz's Law.)

Now reformulate in terms of the net magnetic flux M = B L x through the loop: V = B L dx/dt = - dM/dt (Faraday's Law).

This also works for an arbitrary shaped loop, in which case V is the integral of E around the closed path (loop) enclosing the area through which the magnetic flux is changing.

Implausible Generalizations:

  1. What if you leave the loop still and move the magnet instead? This is just a shift of reference frame, so the Hall voltage shouldn't suddenly disappear. But where does the "induced EMF" come from now? If you use Faraday's formulation it doesn't matter why the flux is changing. Hmmm....

  2. What if you are making B with another coil which you turn on/off? (The basic idea of a transformer!) Again, Faraday's Law gives the right answer "magically".

  3. What if you poke a long straight solenoid through the loop and turn it on? No field acts anywhere around the loop, and yet it magically "knows" that the flux through it has changed!

Faraday's Law is more general than my derivation!

Remember the Order:

  1. What is M?

  2. - Rate of change of M = induced EMF.

  3. Induced EMF causes (if it can) a current to flow (through resistance R, if any).

  4. That current (if it can flow) will (would) make its own magnetic flux to counteract the original flux change. (Lenz's Law)

  5. If real currents flow, magnetic forces result. These are a result of the induced current; include them as the last step in this description.

Some Nice Demos


"Inductance"

by Jess H. Brewer on 2003-03-05:

Inductance

Suppose the long straight solenoid has a cross-sectional area A. Then the magnetic flux through it is M = N A B (since B is uniform inside and each field line links all N turns). Thus M = N A µ0 n I = L I if we define the inductance of the solenoid to be L = A µ0 N2/, where is the actual length of the solenoid. This can also be written L = µ0 n2 A. Note that A is the volume of the inside of the solenoid (where the field is).

Similarly for the toroidal solenoid if it has a rectangular cross section so that integrating B over that area is easy: Ltoroid = (µ0/2) N2h log (b/a), where h is the height of the solenoid and a & b are its inner & outer radii, respectively.

Note that in each case L has the form µ0 N2 x, where x is some length. Thus if L is measured in Henries [1 Henry = 1 Weber per Amp, where a Weber (1 Tesla metre2) is the unit of magnetic flux] then µ0 has units of Henries per metre.

Examples:

First let's ask how big a typical coil's inductance might be. If we make a circular 1000-turn coil 1 cm in radius and 10 cm long (easy enough to make in your kitchen) it would have an inductance of about 4 mH [milliHenries]. Work it out.

Now some demonstrations:

  1. Magnetically damped pendulum and eddy currents.

  2. Transformers and nail melter.

  3. Jumping rings and forces on induced currents.

Stored Energy:

Faraday's Law can be written V = - L dI/dt. If we move a bit of charge dQ = I dt through the wire against that EMF, we do electrical work dUL = - V dQ = L I dI. Integrating from I = 0 up to the final current gives UL = (1/2) L I2.

In a long solenoid, I = B/µ0n and L is given above, so UL = (1/2) (A µ0 n2 ) (B/µ0N)2 = (1/2µ0) B2 A . But A is just the volume of the interior of the solenoid (where the field is), so the energy density per unit volume stored in the solenoid is given by umagn. = (1/2µ0) B2.

Like the analogous result for the energy density stored in an electric field, this result is completely general, far moreso than this example "derivation" justifies.


"Inductance in Circuits"

by Jess H. Brewer on 2003-03-05:

Work through an assortment of "simple" (meaning all elements in series) circuits: (V0 stands for a battery)

Go on from there . . . .

For each case, first picture the Mechanical analogue and then ask "What happens?" before launching into the mathematics.


"AC Circuits"

by Jess H. Brewer on 2003-03-10:

What happens when a series LCR circuit is driven by a sinusoidal voltage at a given frequency? You have already seen the answer in the 109 lab; now let's see if we can understand it better.

The Mechanical Analogue

You have seen this several times before.

The Steady-State Solution

Using complex notation. Current is what we really want to know. We'd like to write Vsource = I Reff. Then Reff = R - i XC + i XL, where XC = 1/C and XL = L are the reactances of the capacitor and inductor, respectively, and is the driving frequency (in radians per second, don't forget!).



"Interference" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"Interference in Films & Slits"

by Jess H. Brewer on 2003-03-24:

Huygens' Principle

"Every point on an advancing wave front may be considered a source of outgoing spherical waves." [paraphrased] The concept of a "wave front" is a little vague, of course; you can think of it as the "crests" of waves if you are visualizing waves in water or on stretched strings, but in 3 dimensional waves a "crest" corresponds to a locus of maximum (positive) amplitude of the wave. In general any locus of fixed phase will do just as well, as long as you use the same fixed phase (plus 2) to defined the adjacent "wave front". Naturally we make no attempt to draw 3D spherical waves on a flat page; all the 2D pictures are meant only as "conceptual shorthand". This will be even more abstract as we start drawing "phasor" diagrams.

Be sure to review your trigonometry - we'll be using it!

Interference from TWO SLITS

(Young's experiment modernized)

The "near field" intensity pattern (where "rays" from the two sources, meeting at a common point, are not even approximately parallel) is difficult to calculate, though it is easy enough to describe how the calculation could be done. We will stay away from this region - far away, so that all the interfering rays may be considered parallel. Then it gets easy!

Simplified sketch assuming incident waves hitting the barrier in phase (i.e. normal incidence) shows an obvious path length difference of = d sin between the waves heading out from the two slits at that angle. If this path length difference is an integer multiple of the wavelength we get constructive interference. This defines the nth Principal Maximum (PM):

d sin n = n

Often we are looking at the position of interference maxima on a distant screen and we want to describe the position x of the nth PM on the screen rather than the angle n from the normal direction. We always define x = 0 to be the position of the central maximum (CM) - i.e. = 0. If the distance L from the slits to the screen is >> d (the distance between the slits), as it almost always is, then we can use the small angle approximations sin tan so that n n /d and xn = L tan n L n giving xn n L /d.

Be sure you can do calculations like these yourself. Such problems are almost always on the final exam.

Time permitting, I will start on Multiple Slit Interference. The handout covers this in detail; if I don't cover it today, be sure to study the handout over the weekend!


"N-Slit Interference"

by Jess H. Brewer on 2003-03-25:

PHASORS ON STUN!

We now take you to a world beyond time and space, a world of pure mathematics where what you see are wave amplitudes and phases of different rays of a coherent wave with a given frequency and wavelength, interfering to make a combined amplitude - the world of The Phasor Zone. (Dew-dew-dew-dew, dew-dew-dew-dew, DEW-diddly-ew...)

In this abstract world each wave is seen as an amplitude Ai pointing away from some origin at a phase angle i in "phase space" - a phasor. All the phasors representing different wave amplitudes are "precessing" about the origin at a common angular frequency (the actual frequency of the waves) but their phase differences do not change with time. Thus we can pick one wave arbitrarily to have zero phase and "freeze frame" to show the angular orientations (and lengths) of all the others relative to it.

Phasors are vectors (albeit in a weird space) and so if they are to be added linearly we can construct a diagram for the resultant by drawing all the amplitudes "tip-to-tail" as for any vector addition. If there are any configurations that "close the polygon" (i.e. bring the tip of the last phasor right back to the tail of the first) then the net amplitude is zero and we have perfect destructive interference!

For an idealized case of N equal-amplitude waves out of phase with their neighbours by an angle we will get a minimum when N = n(2), satisfying the above criterion. This is the condition for the nth minumum of the N-slit interference pattern; we usually only care about the first such minimum, which occurs where N = 2.

To see where in real space that first minimum occurs, we have to go back to the origin of the phase differences due to path length differences: /2 = / = d sin /, giving

sin first min. = /N.


"N-Slit Interference"

by Jess H. Brewer on 2003-03-30:

PHASORS ON STUN!

We now take you to a world beyond time and space, a world of pure mathematics where what you see are wave amplitudes and phases of different rays of a coherent wave with a given frequency and wavelength, interfering to make a combined amplitude - the world of The Phasor Zone. (Dew-dew-dew-dew, dew-dew-dew-dew, DEW-diddly-ew...)

In this abstract world each wave is seen as an amplitude Ai pointing away from some origin at a phase angle i in "phase space" - a phasor. All the phasors representing different wave amplitudes are "precessing" about the origin at a common angular frequency (the actual frequency of the waves) but their phase differences do not change with time. Thus we can pick one wave arbitrarily to have zero phase and "freeze frame" to show the angular orientations (and lengths) of all the others relative to it.

Phasors are vectors (albeit in a weird space) and so if they are to be added linearly we can construct a diagram for the resultant by drawing all the amplitudes "tip-to-tail" as for any vector addition. If there are any configurations that "close the polygon" (i.e. bring the tip of the last phasor right back to the tail of the first) then the net amplitude is zero and we have perfect destructive interference!

For an idealized case of N equal-amplitude waves out of phase with their neighbours by an angle we will get a minimum when N = n(2), satisfying the above criterion. This is the condition for the nth minumum of the N-slit interference pattern; we usually only care about the first such minimum, which occurs where N = 2.

To see where in real space that first minimum occurs, we have to go back to the origin of the phase differences due to path length differences: /2 = / = d sin /, giving

sin first min. = /N.

Analytical Solution for N Slits

You need to actually remember, understand, and be able to use everything above. The full derivation of the formula for the intensity as a function of is another matter. It is shown in gory detail on the Phasors handout and I will go over it in class, but you will not be expected to derive it, nor would there be many occasions when you would need to use it, as long as you can reason out the positions of principal maxima and secondary maxima or minima using the qualitative arguments above. Nevertheless, it is nice to see a real derivation; you may even wish to run the result through Mathematica (or some other software package) to make nice plots of "interference patterns" for your own amusement and the edification of your friends.



"Maxwell's Equations" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Displacement Current"

by Jess H. Brewer on 2003-03-10:

Ampère's Law revisited

What happens when you run a current "through" an uncharged capacitor? Apply Ampère's Law around the wire; now extend the "open surface bounded by the close loop" so that it passes through the gap in the capacitor without cutting any current-carrying wires. (Imagine that you are making a big soap bubble with a hoop.) Does the magnetic field around the loop suddenly disappear? I think not!

Maxwell proposed a "time-varying electric flux" term symmetric to the changing magnetic flux in Faraday's Law to resolve this paradox. Suddenly a time-varing electric field generates a magnetic field, as well as the reverse.


"A Wave in Nothing"

by Jess H. Brewer on 2003-03-16:



"The Magnetic Field" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"I x B: the Lorentz Force"

by Jess H. Brewer on 2003-02-23:

The Magentic Field: why do we bother to define such a thing, rather than just looking at the direct force law? Well, have you ever looked at the direct force law? It's too hard!

Instead we postulate some pre-existing magnetic field B (Who knows where it came from? None of our business, for now.) and ask, "How does it affect a moving charged particle?" The answer (SWOP) is the Lorentz Force:

F = Q(E + v x B)

where v is the vector velocity of the particle, "x" denotes a cross product (review this!) and we have thrown in the Coulomb force due to an electrostatic field E just to make the equation complete.

Note that a bunch of charged particles flowing through a short piece of wire (what we call a current element I d) is interchangeable with a single moving charge Qv. Discuss units briefly.

Speaking of units, the Coulomb is defined as an Ampere-second, and an Ampere is defined as the current which, when flowing down each of two parallel wires exactly 1 m apart, produces a force per unit length of 2x10-7N/m between them. No kidding, that's the official definition. I'm not making this up!

Some Disturbing Examples

Consider two like charges at rest with respect to each other: the force of each on the other is repulsive, right? Now fly over them in a jet plane and look down on them: in your frame they are moving parallel to each other; this gives a small attractive force. If you fly by in the Enterprise instead, you can go much faster; eventually the Coulomb repulsion can be overcome by the magnetic part of the Lorentz force. But surely the particles are either attracted or repelled, not both!

Wait! It gets worse! Now try visualizing the forces between two charged particles moving at right angles to each other. What happened to Newton's Third Law?! This conundrum is only resolved by the relativistic transformations of E and B. (Stay tuned . . . . )

Realistic Numbers

In the lab one can "easily" make a field of 1 T (about 20,000 times the Earth's field here). A wire 1 m long carrying 1 A of current will "feel" a net sideways force of only 1 N. Motors etc. need lots of "turns" of wire to make a decent sized torque.

Circulating Charges

If v is perpendicular to B, we get a very familiar situation: the force on the particle is always normal to its velocity, so it cannot change its speed; and yet it is constantly accelerated. Ring a bell? Come on, you know this: it's gool 'ol uniform circular motion! Solve the familiar equations to get

p = Q B r

where p is the momentum and r is the radius of the orbit. "It turns out" that this relation is relativistically correct, but you needn't concern yourself with this now.

Playing with angular frequency and such reveals the nice feature that the period of the orbit is independent of the speed of the particle! This nice feature (which is not true relativistically, but only at modest speeds) is what makes cyclotrons possible. See TRIUMF.


"What B Do!"

by Jess H. Brewer on 2003-02-25:

More tricks with The Lorentz Force:     F = Q (E + v x B)

Cyclotrons

Since (for E = 0) F is always perpendicular to both v and B, the Lorentz force can never change the speed (or kinetic energy) of the particle. In a uniform B, the resultant motion in the plane perpendicular to B is uniform circular motion with a radius of curvature r given by v2/r = QvB/m or p = mv = QBr. Since v = r , this means = QB/m, a constant angular frequency (and therefore a constant orbital period) regardless of v! Faster particles move in proportionally larger circles so that the time for a full orbit stays the same (as long as v << c). This is what makes cyclotrons possible.

Magnetic Mirrors

In general, a charged particle moves along a spiral in a magnetic field. If it moves toward a region of higher field, its momentum perpendicular to the field increases at the expense of its momentum along the field; eventually the latter stops and reverses direction -- a magnetic mirror!

Examples: the van Allen Belt, Tokomaks and Cosmic Rays from the Universe's biggest accelerators.

Remember: the Lorentz force does no work! (It's like a really smart Physicist. :-)

Wien Filters

If v, B and E are all mutually perpendicular, the particle will pass undeflected if E = vB. This makes a nice velocity selector. If you also measure the radius of curvature of the same particle's path in B with no E, you know its momentum. Putting these together gives you the ratio Q/m. If you know Q (which was not so easy until Milliken's "oil drop" experiment) then you know m. This is the basis for conventional mass spectroscopy. However, the cyclotron is an even better mass spectrometer. Why?

Hall effect

Transverse voltage due to moving charges trying to curve in a magnetic field. Useful for determining both the concentration (number per unit volume) and the magnitude and sign of the charge on individual carriers in any material.

Rail Gun

Just a quick hand-waving description. Go look on the Web for many designs, if you're interested; but spooks will probably be watching you thereafter.


"Do B, do B, do!"

by Jess H. Brewer on 2003-02-28:

Biot & Savart vs. Ampère: now that you know how to do integrals, you're expected to use them! (Dang! Ignorance is easier!)

Law of Biot & Savart

Long Straight Wire via Biot & Savart: Derivation shown for general case of a finite wire segment. From this one could fairly easily find the field produced by a loop made of several straight segments. Answer: field makes right-handed circles around the wire, magnitude (for a long wire) B(r) = µ0I/2r where r is the distance from the wire. (Simple answer fron a very complicated derivation.)

Circular Current Loop via Biot & Savart: too hard to calculate the field anywhere except on the axis of the loop. There (by symmetry) the field can only point along the axis, in a direction given by the RHR: curl the fingers of your right hand around the loop in the direction the current flows, and your thumb will point in the direction of the resulting magnetic field. (Sort of like the loops of B around a line of I, except here B and I have traded places.) As usual, symmetry plays the crucial role: current elements on opposite sides of the loop cancel out each other's transverse field components, but the parallel (to the axis) components all add together. As for the electrostatic field due to a ring of charge, we get the same contribution to this non-canceling axial field from each element of the ring.

Ampère's Law

The integral of B// dl around a closed loop (where B// is the component of B along the path at each element dl) is equal to µ0 times the net current Iencl linking the loop (i.e. passing through it).

(Used like Gauss' Law only with a path integral.)

Long Straight Wire via Ampère's Law: It's so easy!

Any Cylindrically Symmetric current distribution gives the same result outside the conductor; inside we get an increase of B with distance from the centre, reminiscent of Gauss' Law....

Circular Current Loop via Ampère's Law: Forget it! Ampère's Law is of no use unless you can find a path around which B is constant and parallel to the path. There is no such path here.



"Thermal Physics" Topic

Found 6 Lectures on Mon 07 Oct 2024.

"First Class!"

by Jess H. Brewer on 2003-01-04:

WHAT IS PHYSICS ABOUT?

Until about the 16th Century, science was dominated by the Aristotelian paradigm, caricatured as follows: "Get to know how things are." That is, concentrate on the phenomena; for instance, what happens when you touch a hot stove? If you asked an Aristotelian why your finger gets uncomfortably hot, the answer would be, "Because that's the way it works, stupid." We are still pretty Aristotelian in our hearts today; the textbook reflects this -- it generally delivers a concise description of "how things are" (usually as a concise formula in terms of defined quantites) and then shows how to use that principle to calculate stuff; only later (if ever) does it show why the world behaves that way. Starting nominally with Galileo, "modern scientists" began to ask questions that Aristotelians whould have considered impertinent and even arrogant, like, "Why does the heat flow the way it does?" or "How heavy is an atom?" or "Why are there only three Generations?" [The last refers to leptons and quarks, not Star Trek.]

In my opinion, PHYSICS is about those impertinent questions. It goes like this: we observe a PHEMONENON and gather empirical information about it; then we MAKE UP A THEORY for why this behaviour occurs, DERIVING it mathematically so we can check it for consistency, extend it and finally use it to PREDICT hitherto unobserved NEW phenomena as well as answering our original questions. Then we can go do EXPERIMENTS to see if the predicted phenomena do in fact occur. If not (usually), back to the drawing board. But over time, this has given us a ladder to climb....

I am going to try to follow this sequence in my lectures, so that PHYS 108 will have some of the flavour of actual science as you will experience it if you become a Scientist (not just Physicists). Some of you won't like it. Sorry. As one of the lesser philosophers of the 20th Century said, "You can't please everyone, so you have to please yourself." And speaking of songs...

(musical introduction to Thermodynamics)

I mainly want everyone to understand that the approach I am taking to introducing Thermal Physics is very unconventional, and that the glib nonsense you were probably taught in high school is not what I expect you to understand by entropy or temperature.


"Entropy and Temperature"

by Jess H. Brewer on 2003-01-08:

PHENOMENON: If you touch a hot stove, you get burned.
QUESTION: How come?
HYPOTHESIS: Energy flows spontaneously from the hot stove to the cooler skin because of random exchanges.
THEORY: Wait a minute, I have more questions! What does "hot" mean? "Cooler"? What sort of "random exchanges"? In any case the so-called "hypothesis" just begs the question of "Why?" -- we need to start earlier.
Revised QUESTION: What's "hot"?
HYPOTHESIS: It has something to do with randomness and energy.
THEORY: Let's make up the simplest possible definition of what we mean by "random": the Fundamental Assumption of Statistical Mechanics, namely

Every accessible fully specified state of the system is a priori equally likely.

Whoa! This has some ringers in it. We need to define (as well as we can) exactly what we mean by "accessible", "fully specified state" (or for that matter "state"), "system" and "a priori". Here we get to the details, for which I think it is appropriate to say, "You had to be there!" Some topics touched upon: Dirac notation ("|a>"), energy conservation, parking lots, counting, binomial distributions, the multiplicity function, entropy, microcanonical ensembles, maximum likelihood, extrema and derivatives, temperature and the Cuban economy.

Everything is on the Thermal Physics handout (Ch. 15 of the Skeptic's Guide). I went from the beginning through the definition of (the dimensionless form of) entropy and on to the definition of inverse temperature as the criterion for the most probable configuration, i.e. thermal equilibrium. This stuff is essential, fundamental and important! I expect you to know it well enough to reproduce the derivation on an exam. (Not many things fall into this category.) Note that the derivative of entropy with respect to energy is the inverse temperature; thus when entropy is a dimensionless number, temperature is measured in energy units.


"Hot, Hotter and Boltzmann"

by Jess H. Brewer on 2003-01-08:

  1. So What IS "Hot" Anyway?
    "It takes two to tango." Put two systems in contact so they can exchange energy (U). Now every accessible microstate of the combined system is equally likely; but there are more such states in some "configurations" (divisions of U between the two systems) than others. If U1 is the total energy in system 1 and U2 is the total energy in system 2, U1 + U2 = U is a conserved constant and any increase in U1 implies a correspinging decrease in U2:

    dU2 = - dU1

    Does such a change lead to more overall possibilities? For a given configuration, the net multiplicity is the product of the multiplicities of the individual systems, so the net entropy is the sum of the entropies of the individual systems. If the net entropy increases when we take dU1 out of system 2 and move it into system 1, then this new configuration is more likely -- i.e. such an energy transfer will happen spontaneously. This is exactly what we mean when we say that system 2 is hotter than system 1! To turn this into a formal definition of temperature we need some mathematics.

    .

    .

    .

  2. A Model System: N spins in a magnetic field.
    Energy per spin = plus or minus µB where µ is the magnetic moment of one spin.
    Total energy U depends only on B and the number n of spins up. (The rest are down.)

    U = (n - [N - n]) µB = (2n - N)µB


    Thus the multiplicity function is a binomial distribution, which is approximately Gaussian for large N.
    Entropy (log of a Gaussian) is an inverted parabola.
    Slope of entropy vs. energy goes through zero and then goes negative. Therefore temperature goes to infinity, then jumps discontinuously to and finally approaches zero from the negative side. What does this mean?

  3. Really Small Systems: a single degree of freedom Put such an infinitesimal "fully specified state" in contact with a large "heat reservoir" at temperature T. This is called a Canonical Ensemble. What can we say about the probability of finding the tiny "system" in that particular fully specified state?

    .

    .

    .

    Mathematical derivation of the Boltzmann Distribution.

    Note that the probability must be normalized.


"Particle in a Box"

by Jess H. Brewer on 2003-01-15:

Discussion of standing waves, http://musr.physics.ubc.ca/~jess/hr/skept/Waves/node9.html> quantization and de Broglie's Principle:

= h/p

[For an introduction to Quantum Mechanics in the form of the script to a comical play, see The Dreams Stuff is Made Of (Science 1, 2000).]
.

.

.

Discrete wavelengths, momenta and energies. Lowest possible energy is not zero. As the box gets smaller, the energy goes up!

Handwaving reference to black holes, relativistic kinematics, mass-energy equivalence and how the energy of confinement can get big enough to make a black hole out of even a photon if it is confined to a small enough region (Planck length).


" Kinetic Theory of Gases"

by Jess H. Brewer on 2003-01-15:

EQUIPARTITION OF ENERGY: IDEAL GAS:


"Momentum Space"

by Jess H. Brewer on 2003-01-17:

Allowed states are evenly spaced in momentum but not in energy, which is what we want in our Boltzmann distribution. Since p E, we expect (E) 1 / E. (Sketch.)

Moving to 3-D picture, there is one allowed state (mode) per unit "volume" in p-space. But if what we want is the density of states per unit magnitude of the (vector) momentum, there is a spherical shell of "radius" p and thickness dp containing a uniform "density" of allowed momenta whose magnitudes are within dp of p. This shell has a "volume" proportional to p2 and so the density of allowed states per unit magnitude of p increases as p2. This changes everything!

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.

.

The details are on the Momentum Space handout. You may feel this is going too far for a First Year course, and I have considerable sympathy for that point of view. I simply wanted you to have some idea why the Maxwellian energy and speed distributions have those "extra" factors of E and v2 in them (in addition to the Boltzmann factor itself, which makes perfect sense). The textbook (perhaps wisely) simply gives the result, which is too Aristotelian for us, right?

Rest assured that I will not ask you to reproduce any of these manipulations on any exam. At most, I will ask a short question to test whether you understand that one must account not only for the probability of a given state being occupied in thermal equilibrium (the Boltzmann factor) but also how many such states there are per unit momentum or energy (the density of states) when you want to find a distribution.



"Waves" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Wave Review"

by Jess H. Brewer on 2003-03-18:

Today I want to spend some time reviewing the general properties and behaviour of waves. Only a few of the topics will be new, but for the rest of the course I am going to be relying on your deep and intuitive understanding of how waves behave, so I will not just rely on what you learned last term.

The Electromagnetic Spectrum

. . . from ~1 Hz seismic waves (wavelength ~108 m) to ~1020 Hz gamma rays (wavelength ~10-12 m). We will, out of human biological chauvinism, pay most attention to the visible spectrum between ~400 and ~800 nm in wavelength.

Simple Harmonic Motion (SHM) in Time and Space

. . . a review of sinusoidal travelling waves.

Solutions of the Wave Equation

. . . the linear Wave Equation has solutions that are not sinusoidal. In fact, any well-behaved function of only u = x - ct, where c is the wave's propagation velocity, will automatically satisfy the Wave Equation. Same for u = x + ct, but this describes a wave propagation in the negative x direction.

Which way is it going? To see the answer, pick a point of well defined phase on the wave (for instance, where it crosses the x axis) and then let t increase by a small amount dt. This changes the phase; what would you need to do with x to make the phase go back to its original value? If adding dx to x would compensate for the shift in t, then the wave must be moving in the positive x direction. If you must subtract dx from x to get this effect, it is moving in the negative x direction. Be sure you understand this thoroughly.

Actual Wave Functions: Plane and Spherical Waves

The standard "plane wave" propagating in the z direction can be generalized to propagate in the k direction, where k is called the wave vector. It has the same magnitude as usual, k = 2/, but the scalar kz is replaced by the dot product kr (where r is the vector position where we want to know the wave's amplitude). Imagine the wave "crests" as plane sheets stretching off to infinity in both directions perpendicular to k, marching along in the k direction at c. Obviously the plane wave is an idealization. We won't use this formulation explicitly very often, but it serves to remind us that the wave has a well defined direction of propagation, which we habitually express in the form of rays, a picture inherited from Newton, who insisted that light was particles following trajectories like little billiard balls, until Huygens showed that it was indeed waves. (We now know they were both right!)


"Reflection, Refraction & Interference"

by Jess H. Brewer on 2003-03-21:

Group vs. Phase Velocity

For a solution of "the" Wave Equation, they are the same thing: if = c k, then vg = d/dk = c = /k = vp

But there are lots of other wave equations (a good example being the Schroedinger Equation for "matter waves" which you will encounter next year if you take Physics 200) which do not have this simple linear relationship between the frequency and the wavelength. We will not dwell on this in P108, but you should be aware that actual information (or matter itself, in the case of matter waves) moves at the group velocity vg = d/dk, not at the phase velocity vp = /k.

Using "Rays"

Reflection: phase reverses ( = 2) at reflection from a denser medium with a larger index of refraction (like for a rope attached to the wall); phase does
not
reverse at reflection from a less dense medium with a smaller index of refraction (like for a rope with a dangling end).

Refraction: "slow light" -- index of refraction n = cvacuum/c'medium (always 1 or greater).

Snell's Law: n sin = n' sin '.

Total Internal Reflection (Ltd.)

"Effective path length"

INTERFERENCE: "Beats" in Space

This applies only for waves with the same frequency.

Disturbances of a linear medium just add together. Thus if one wave is consistently "up" when the other is "down" (i.e. they are "180o out of phase") then the resultant amplitude at that position is zero. This is called "destructive interference". If they are both "up" (or "down") at the same time in the same place, that's "constructive interference".

Thin Films

Assuming normal incidence, add together the "rays" reflected from both surfaces of the film. Remember the phase change at any reflections from denser media. Then add in the phase difference = 2 (/) due to the path length difference and you have the net phase difference between the two reflected waves. When this is an integer multiple of 2 you have constructive interference. When it is an odd multiple of , you have destructive interference. That's really the whole story.

Examples: the "quarter wave plate" and the soap film. Oil on water and the fish poem.



"Weird Science" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Weird Science"

by Jess H. Brewer on 2003-01-19:


"Vector Fields"

by Jess H. Brewer on 2003-01-22:

"The force that would be there" - Electric Field

Vector fields and visualization.

Simple problems with point charges. Superposition of electric fields from different sources - just add 'em up (vectorially)!

Not so simple problems: continuous charge distributions.

Example: the electric field on axis due to a ring of charge can only be calculated by "brute force" integrating Coulomb's Law. Fortunately it is quite easy, as long as we stay on the axis where transverse components cancel by symmetry.

Slightly harder: the electric field on axis due to a disc of charge is the sum of the fields from all the little rings that make up the disc.

Always check that the result you calculate behaves as expected (namely, Coulomb's Law) as you get so far away from the charged object that it looks like a point charge.




"UBC Physics 473" Course


"UBC Physics 107 [Fall 2004]" Course

"Emergence of Mechanics" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Mathematics as the Agent of Emergence"

by Jess H. Brewer on 2003-12-16:



"Introduction" Topic

Found 1 Lectures on Mon 07 Oct 2024.

""

by junaid on 2006-06-01:



"Rigid Body Motion" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Moments of Inertia"

by Jess H. Brewer on 2004-10-06:

Definition of a Rigid Body: a system of particles in which the distance between any two particles is held fixed.

Possible motions of a rigid body:

  1. Translational motion of the centre of mass (CM) as if all the mass were located there and all external forces acted thereon.
  2. Rotational motion about an axis through the CM.
Of course one can always have rotation about some other axis not through the CM, but then one has to take into account the rotational motion of the CM as well. More on this later.

Inertial factors: we are used to m being a fixed, scalar property of a particle, determining both how much F it takes to produce any a and how much p we get for a given v. In the same way, there is a measure of rotational inertia called the moment of inertia IA about axis A that tells us how much torque it takes to produce a given angular acceleration and how much angular momentum we get for a given angular velocity.

Show for an arbitrary axis through an arbitrary rigid body that the moment of inertia about that axis is the sum (integral) of the square of the perpendicular distance from the axis times the element of mass at that distance.

Thus the moment of inertia of a hoop of mass M and radius R about a perpendicular axis through its centre is ICM = MR2. The same goes for a cylindrical shell. More examples on Friday.


"Rockin' Rolling"

by Jess H. Brewer on 2004-10-08:

What about the moment of inertia about some other axis?

Parallel Axis Theorem

If we know the moment of inertia ICM of an arbitrary rigid body about an axis through its CM, it can easily be shown that the moment of inertia about a different axis A parallel to the CM axis but a perpendicular distance h away from it is given by

IA = ICM + M h2.

Perpendicular Axis Theorem

(Applies only to thin flat plates.) Pick any point on the plate; draw the z axis through that point perpendicular to the plate and the x and y axes in the plane of the plate. The distance r of a given mass element from the z axis obeys r2 = x2 + y2, where x and y are its distances from the y and x axes, respectively. Thus

Iz = Ix + Iy.

Some Examples

A thin Rod of length L

A Rectangular Plate of length Lx and width Ly

Here the integrals for Ix and Iy are the same as for a thin rod: Ix = (1/12)M Lx2 and Iy = (1/12)M Ly2. Applying the Perpendicular Axis Theorem gives

Iz = (1/12)M (Lx2 + Ly2).

A uniform Disc

...is a collection of hoops, each with its own radius r and its own mass (proportional to r). Since moments of inertia add, we just sum (integrate) over all the hoops to get (with only a few steps)

ICM = (1/2)M R2.

A Spherical Shell

...can also be built up out of hoops, each of which is centred on the same perpendicular axis through its centre. The straightforward integration gives

ICM = (2/3)M R2.

A Solid Sphere

...can be built up out of hoops or discs, each of which is again centred on the same perpendicular axis through its centre. The slightly more challenging integration gives

ICM = (2/5)M R2.

Each of these takes a little while to calculate, and the only difference between them is the numerical factor out in front of the M R2 or M L2 or whatever.

Although you can do the calculation yourself using only simple integrations and the two Theorems described above, this is one of those few cases where it is a good idea to just memorize the numerical factors that go with the different common shapes, to save yourself time and energy on homework and exams.

Kinetic Energy of Rotation

An easy derivation gives

K = (1/2)M V2 + (1/2) ICM 2

where V is the velocity of the CM.

Rolling Motion of a Wheel

Demonstration of two apparently identical cylinders rolling down an inclined plane: one reaches the bottom significantly ahead of the other. Why?

(Several theories proposed; vote taken on which was wrong.)

Explanation: In general, the angular motion is independent of the translational motion. But in the case of rolling without slipping, the position on the surface is locked to the angle through which the wheel has turned, and so likewise the speed parallel to the plane and the angular velocity of rolling:

v = R .

Applying this to the net kinetic energy, which must equal the gravitational potential energy lost as the wheel rolls downhill, we find that the smaller the moment of inertia per unit mass, the larger the velocity at the bottom of the slope.




"UBC Physics 108 [Spring 2005]" Course

"DC Circuits" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"Batteries, Resistors and DC Circuits"

by Jess H. Brewer on 2005-02-09:

We have explained why the "voltage drop" across a capacitor is -Q/C. Think of the capacitor as a rubber balloon which you can fill with a charged "fluid"; the more fluid you inject, the harder it tries to squirt it back out. The next circuit element to consider is the battery, whose voltage "drop" is +Vo. You can think of the battery as a reservoir full of charged fluid that is stored at higher elevation and therefore gives you constant pressure in the pipes. Which brings us to the next circuit element: the "pipe" through which the fluid must flow against friction; this is the resistor. The voltage drop for a resistor is -IR. This can be understood in microscopic detail, but for now let's lean heavily on analogy:

Adding Resistances: in series (just add 'em up!) and in parallel (add inverses to get equivalent inverse).

Kirchhoff's Rules:

  1. Charge is conserved. Whatever current goes into a junction must also come out. Net charge on any isolated part of the circuit is zero.

  2. Potential is single valued. A trip around any closed loop must get you back to the same voltage you started with. ("Sum of the voltage drops equals zero.")

Now you understand both C and R. Let's put them together.

RC is a Time Constant

Consider that R is measured in Ohms = Volts/Amp = Volt-sec/Coul whereas C is measured in Coul/Volt; thus RC is measured in seconds. This simple dimensional analysis should make you suspect that there is a time constant that grows with both C (charge capacity) and R (resistance to the flow of charge).

Discharging a Capacitor Through a Resistor

Start with an open circuit with a charged capacitor connected to a resistor; now close the switch. What happens? Well, basically, the charge is going to bleed off the capacitor through the resistor. The bigger the capacitor, the more charge there is to bleed off (for a given initial voltage drop across the capacitor) and the bigger the resistor, the slower it bleeds. The voltage on the capacitor is propostional to its remaining charge and the voltage drop across the resistance (proportional to the current, which is the rate of change of the charge on the capacitor) is exactly balanced by the voltage on the capacitor (because potential is single-valued, see Kirchhoff's Rule #2). So you pretty much know just what to expect; it is just like a projectile slowing down under viscous flow: exponential decay with a mean lifetime = RC.

Now do it mathematically: solve the differential equation for Q(t).

Charging Up a Capacitor Through a Resistor with a Battery

Start with the capacitor uncharged and then apply a constant voltage through a resistor. What happens?


"Resistance is Futile!"

by Jess H. Brewer on 2005-02-10:

Drude Theory

Ski Slope Analogy:Idealized skis: frictionless. Idealized skiers: indestructible morons. Idealized collisions: instantaneous, perfectly inelastic.

Drift velocity vd = acceleration (eE/m) mean time between scattering collisions .

Problem: how can be independent of the "steepness of the slope"? Short excursion into Quantum Mechanics and Fermi-Dirac statistics: electrons are fermions (half-integer spin) so no two electrons can be in the same state. Thus the lowest energy states are all full and the last electrons to go into a metal occupy states with huge kinetic energies (for an electron) comparable to 1 eV or 10,000 K. Only the electrons at this "Fermi level" can change their states, so only they count in conduction. So our ideal skiers actually have rocket-propelled skis and are randomly slamming into trees (and each other) at orbital velocities (we will neglect the problems of air friction); the tiny accumulated drift downhill is imperceptible but it accounts for all conduction.

J = "flux" of charge = current per unit perpendicular area (show a "slab" of drifting charge) so J = n e vd, where n is number of charge carriers per unit volume. (For Cu, n is about 1029 m-3.)

Ohm's Law

J = E, defining the conductivity = n e2 /m, measured in Siemens per metre (S/m) if you like SI units (1 S = 1 A/V). I don't.

For Cu, is around 108 S/m. Putting this together with n, e and me = 9 x 10-31 kg, we get ~ 10-13 s. At vF ~ 106 m/s this implies a mean free path ~ 10-7 m. Compare lattice spacing ~ 10-10 m. The drift velocity vd is only ~ 10-3 m/s.

In semiconductors, n is a factor of 10-7 smaller and vd is a factor of 107 larger, almost as big as vF! So in some very pure semiconductors transport is almost "ballistic", especially when the size of the device is less than .

Briefly discuss superconductors.

The inverse of the conductivity is the resistivity , measured in Ohm-meters. (1 Ohm = 1 V/A).

Use a cartoon of a cylindrical resistor of length L and cross-sectional area A to remind how this works, giving R = L/A and the familiar V = I R.


"RC with AC & Introduction to B"

by Jess H. Brewer on 2005-02-16:

See the PDF file on AC RC Circuits for all the details I covered in class (and then some).

The Magentic Field

Why do we bother to define such a thing, rather than just looking at the direct force law? Well, have you ever looked at the direct force law? It's too hard!

Instead we postulate some pre-existing magnetic field B (Who knows where it came from? None of our business, for now.) and ask, "How does it affect a moving charged particle?" The answer (SWOP) is the Lorentz Force:

F = Q(E + v x B)

where v is the vector velocity of the particle, "x" denotes a cross product (review this!) and we have thrown in the Coulomb force due to an electrostatic field E just to make the equation complete.

Note that a bunch of charged particles flowing through a short piece of wire (what we call a current element I d) is interchangeable with a single moving charge Qv.

Some Disturbing Examples

Consider two like charges at rest with respect to each other: the force of each on the other is repulsive, right? Now fly over them in a jet plane and look down on them: in your frame they are moving parallel to each other; this gives a small attractive force. If you fly by in the Enterprise instead, you can go much faster; eventually the Coulomb repulsion can be overcome by the magnetic part of the Lorentz force. But surely the particles are either attracted or repelled, not both!

Wait! It gets worse! Now try visualizing the forces between two charged particles moving at right angles to each other. What happened to Newton's Third Law?! This conundrum is only resolved by the relativistic transformations of E and B. (Stay tuned . . . . )

Realistic Numbers

In the lab one can "easily" make a field of 1 T (about 20,000 times the Earth's field here). A wire 1 m long carrying 1 A of current will "feel" a net sideways force of only 1 N. Motors etc. need lots of "turns" of wire to make a decent sized torque.



"Diffraction" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"An Infinite Number of Slits?"

by Jess H. Brewer on 2005-03-26:

Features of Single Slit Diffraction

  1. At the central maximum ( = = 0) One sees the full I0. (Use l'Hospital's rule on (sin x)/x as x goes to zero.)

  2. The intensity goes to zero at any nonzero for which sin = 0, i.e. any integer multiple of . The first minimum of the diffraction pattern occurs when = , which in turn implies

    a sin 1 = .

  3. The secondary maxima of the diffraction pattern can be found by setting the derivative of I with respect to equal to zero (condition for an extremum), giving

    = tan .

    This transcendental equation can be solved by plotting both x and sin x on the same graph and looking for intersections. Don't go looking for an analytical solution.

Circular Apertures

We've been talking about "slits" as if all diffraction problems were one-dimensional. In reality, the most common type is circular, such as telescopes, laser cannons and the pupil of your eye. The following handwaving logic is not a proof, but a plausibility argument:

The narrower the slit, the wider the diffraction pattern. Picture a circular aperture as a square aperture with the "corners chopped off": on average, it is narrower than the original square whose side was equal to the circle's diameter. Thus you would expect it to produce a wider diffraction pattern. It does! The numerical difference is a factor of 1.22:

a sin 1 = 1.22 .


"Dispersion"

by Jess H. Brewer on 2005-04-04:

Dispersion

The wavelength (colour) dependence of the interference pattern from a grating determines how useful it will be for resolving sharp "lines" (light of specific wavelengths) in a mixed spectrum. The rate of change of the angle of the mth principal maximum with respect to the wavelength is called the dispersion Dm of the grating. This is easily shown to have the value

Dm = dm/d = m/d cos m .

Note that there is a different dispersion for each principal maximum. Which m values will give bigger dispersions? Why does this "improvement" eventually have diminishing returns?

Resolving Power

A separate question is How close together () can two colours be and still be resolved by the grating?

Well, the two lines will just be resolved when the mth order principal maximum of one falls on top of the first minimum beyond the mth order principal maximum of the other.

By requiring the path length difference between adjacent slits to differ (for the two colours) by /N (where N is the number of slits) we ensure that the phasor diagram for the second colour will just close (giving a minimum) when that of the first colour is a principal maximum. This gives a resolving power

Rm = / = m N .



"Electrostatics" Topic

Found 5 Lectures on Mon 07 Oct 2024.

"The Electric Field"

by Jess H. Brewer on 2005-01-19:

"The force that would be there" - Electric Field

Vector fields and visualization.

Simple problems with point charges. Superposition of electric fields from different sources - just add 'em up (vectorially)!

Not so simple problems: continuous charge distributions.

Example: the electric field on axis due to a ring of charge can only be calculated by "brute force" integrating Coulomb's Law. Fortunately it is quite easy, as long as we stay on the axis where transverse components cancel by symmetry.

Slightly harder: the electric field on axis due to a disc of charge is the sum of the fields from all the little rings that make up the disc.

Always check that the result you calculate behaves as expected (namely, Coulomb's Law) as you get so far away from the charged object that it looks like a point charge.

PDF or printer-friendly gzipped PostScript files


"Doing Electrostatics the Hard Way"

by Jess H. Brewer on 2005-01-24:

Calculate the torque on an electric dipole in a uniform external electric field. From that, calculate the potential energy of the same dipole in the same field, as a function of its orientation.

Then move on to a hard (but not impossible) problem: the electric field due to a Finite Rod of Charge. See also the usual PDF and printer-friendly gzipped PostScript files.


"Electrostatic Potential"

by Jess H. Brewer on 2005-01-31:

Electrostatic Potential

Notation: I will use V here instead of ["phi"] (chosen in class) because HTML still has no Greek letters except "µ". I can get away with this on the computer because the math symbol V (for potential) is italicized while the abbreviation "V" (for "Volts") is not; so I can write "V = 4 V" without ambiguity.

In principle, it's easier to find E from V (using E = - ) than vice versa, because it's a lot easier to integrate up a scalar function than a vector one! (And derivatives are easy, right?) However, in practice (at the level of P108) we are not going to be evaluating arbitrary, asymmetric charge distributions, but only the simple symmetric shapes and combinations thereof (using the principle of additive superposition). In these cases Gauss' Law allows us to find E easily and find V by simple integrations; so that's mostly what we do.

Examples

(Note: to get the sign right, always check your result against common sense: for a unit [positive] test charge, "uphill" is from negatively charged regions to positively charged regions, so the potential should increase in that direction.)


"Capacitance"

by Jess H. Brewer on 2005-02-04:

Thinking of an isolated conductor as a capacitor is very irregular. We usually think of a capacitor as two conductors with equal and opposite charges, and calculate the capacitance "between" them. Consider for example two perpendicular wires that don't touch. Even if the wires are infinite, their mutual capacitance is finite. (This would be a real challenge to calculate!) I offer a $10 prize to anyone who correctly calculates the capacitance. An extra $5 for the charge distribution along each wire.

Dielectric Materialism (Ch. 29)

Basically just replace by = (where is the dielectric constant, a pure number always 1) and everything takes care of itself. Thus C always gets bigger (by a factor of ) when there is a dielectric in between the plates. [Explain.]

For isotropic, cylindrical and planar geometries, show how potential is calculated from the electric field and how capacitance is in turn calculated from that. See PDF or printer-friendly gzipped PostScript files.


"Electrostatic Springs and Energy Storage"

by Jess H. Brewer on 2005-02-05:

Capacitor as an Electrostatic "Spring"

If you like you can think of 1/C as a sort of "electrical spring constant": if you move Q away from its equilibrium value (zero) you get a "linear restoring voltage".

Arrays of Capacitors

An arbitrary network of capacitors can always be replaced by a single equivalent capacitor.

An array of capacitors in parallel has an equivalent capacitance equal to the sum of their separate capacitances. [Explain.]

An array of capacitors in series has an equivalent inverse capacitance equal to the sum of their separate inverse capacitances. [Explain.]

Electrostatic Energy Storage

Recall the question at the beginning: why isn't a big capacitor a good replacement for a battery? Because the voltage decreases with the remaining charge! This has other implications as well....

The energy required to put a charge Q on a capacitor C is not just VQ! The first bit of charge goes on at zero voltage (no work) and the voltage (work per unit charge added) increases linearly with Q as the charge piles up: V = (1/C) Q. Thus dU = (1/C) Q dQ. Integrating yields U = (1/2C) Q2 or U = (1/2)C V2.

For a parallel plate capacitor, V = E d and C = A / d. Thus U = (1/2)AE2 d. But A d is the volume of the interior of the capacitor (the only place where the electric field is nonzero). Thus if u is defined to be the energy density per unit volume, then we have u = (1/2) E2. "It turns out" that this prescription is completely general! Wherever there is an electric field, energy is stored at a density u given by the formula above.

It is now getting really tempting to think of E as something "real", not just a mathematical abstraction.



"Elementary Particles" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Small Stuff"

by Jess H. Brewer on 2005-04-08:

I gave the cartoon version in class; colorization of the images is not yet complete, but you can download the PDF or printer-friendly gzipped PostScript version and see if it has been updated.



"EXAM" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"First Midterm"

by Jess H. Brewer on 2005-02-09:


"Second Midterm"

by Jess H. Brewer on 2005-03-13:


"Final Exam"

by Jess H. Brewer on 2005-04-08:



"Faraday & Inductance" Topic

Found 0 Lectures on Mon 07 Oct 2024.

"Gauss' Law" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Conservation, Flux & Symmetry"

by Jess H. Brewer on 2005-01-26:

Divisible Conservation

Without some quantity (real or imagined) that is (a) divisible into smaller parts and (b) conserved overall, Gauss' Law has no meaning. Some examples would be the amount of water in a river, the amount of energy in a closed system, the number of dollars in circulation (neglecting reissues, recalls, counterfeiting, lighting cigars with $20 bills, bills used as bookmarks and forgotten, and of course ignoring the real value of a dollar), the amount of light emitted by a source, the smell emitted by a rabbit or the electric "field lines" emitted by a positive charge. In each case, anywhere there is a "source" or a "sink" of the conserved "stuff" has to be accounted for explicitly; the "stuff" does not just disappear or appear out of nowhere without a good reason. But (in order to be interesting in this context) the "stuff" does have to be divisible so that we don't just have one clump of it rattling around. So a quantized thing like the charge on an electron, while conserved, would not be a good candidate; there is a version of Gauss' Law for charge itself, but in Classical Electrodynamics we always deal with such large numbers of electrons etc. that we can get away with pretending that charge is an infinitely subdividable "fluid" just the way we do with water and air.

Flux

Next we need some sort of distributed "motion". If everything is static there is no need for Gauss' Law. The water flows, the energy is exchanged, the money changes hands in commerce, the rabbit smell drifts in the air, the light travels away from its source at c and the electric field "lines" may be thought of (quite accurately, it turns out) as "rays" of zero-frequency light also moving away from the positive-charge source (or falling into the "sink" of a negative charge) at the speed of light. This is a nontrivial thing to visualize, as the "stuff" can be everywhere in different amounts, moving in different directions at different speeds. We have to invent the notion of flux of "stuff", a space-filling vector field that has at each point in space both direction and magnitude (usually in units of "stuff" per unit time per unit perpendicular area). I use a "directed tennis racket" to illustrate how flux can be measured at a given position; this requires an additional new idea: the directed area element.

Gauss' Law

We are now ready to state Gauss' Law in its most primitive form: When some "stuff" leaves a region, there is that much less "stuff" in that region. Doh! Conversely, When some "stuff" enters a region, there is that much more "stuff" in that region. Well, duh! But that's all there is to it. We can use elegant mathematical language to express the same simple idea, but it doesn't make it any less simple. Remember that.

Spreading Out = Thining Out

Here's the part any good hunting dog understands: if the "stuff" (in this case, rabbit smell, whose flux we can think of as "lines of rabbit") is coming from a localized source and spreading out into a larger region, then because the total flux of stuff out through any closed surface is constant (conserved), the flux has to get "thinner" (the smell has to get fainter) as you get further away from the rabbit. This allows a simple routine to consistently bring you closer and closer to the rabbit. [Describe.]

Symmetry

OK, this works. But can we use it to calculate electric fields due to collections of charges? We may be able to say lots of qualitative things just from Homer Simpson logic, but if we want precise, quantitative results we can do nothing with Gauss' Law unless we have symmetry on our side. If the electric field is different everywhere on the Gaussian surface in question, and passes through the surface at various angles in different places, we have not gained a thing by stating the laws of electrostatics in this form. We need to be able to define a simple surface of familiar geometry where the electric field must be both constant and everywhere normal to the surface before Gauss' Law is going to do us a bit of good. But if we can find such a surface, that that big scary-looking surface integral is just E times the total area A through which E passes at normal incidence. In that case we can write Gauss' Law for Electrostatics in the simple form

A D = Qenclosed

where for additional simplicity we have defined D = E.


"Gauss' Law in Action"

by Jess H. Brewer on 2005-01-28:

Continue discussion of cylindrical symmetry. Then do planar symmetry.

If time permits, begin discussion of conductors.



"Interference" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"Adding Amplitudes"

by Jess H. Brewer on 2005-03-22:

Linear Superposition (Adding Amplitudes)

The most remarkable feature of a "linear medium" (including vacuum in the case of electromagnetic waves) is that the amplitude of one wave and that of another are independent of each other -- waves can "pass through" each other without scattering; they just keep going and come out the other side as if the other wave hadn't been there! Moreover, while they are passing through the same region, their amplitudes simply add together, so that when their "crests" or "troughs" coincide, the net effect is a bigger wave, but when the "crests" of one wave coincide with the "troughs" of another, the net effect is a cancellation. This can lead to very complicated behaviour called "interference". Our goal is to find simple tricks to make it seem less complicated, but one should never lose sight of the beautiful, intricate patterns created by interference.

Standing Waves: The most familiar example to players of stringed instruments is probably the case of two waves of equal amplitude, wavelength and frequency propagating in opposite directions (which can be represented mathematically by giving either k or [but not both] opposite signs for the two waves). In this case we get a wave which no longer "travels" but simply "oscillates in place" with nodes where no motion ever occurs. The "particle in a box" example shares with the closed organ pipe and the guitar string the feature that there must be nodes at the ends of the box/pipe/string, a feature that forces quantization of modes even for classical waves.

Beats - Interference in Time: If two waves pass the same location in space (your ear, for instance) with slightly different frequencies then they drift slowly into and out of phase, resulting in a sound of the average frequency whose average amplitude (or its square, the intensity) oscillates at a frequency equal to the difference between the two original frequencies. This is a handy method for tuning guitar strings: as their frequencies of vibration get closer together, the beat frequency gets slower, until it disappears entirely when they are exactly in tune.

Interference in Space: This applies only for waves with the same frequency. Consider two waves of equal amplitude: If one wave is consistently "up" when the other is "down" (i.e. they are "180o out of phase") then the resultant amplitude at that position is zero. This is called "destructive interference". If they are both "up" (or "down") at the same time in the same place, that's "constructive interference".

Thin Films: Assuming normal incidence, add together the "rays" reflected from both surfaces of the film. Remember the phase change at any reflections from denser media. Then add in the phase difference = 2 (/) due to the path length difference and you have the net phase difference between the two reflected waves. When this is an integer multiple of 2 you have constructive interference. When it is an odd multiple of , you have destructive interference. That's really the whole story.

Examples: the "quarter wave plate" and the soap film. Oil on water and the fish poem.


"Two Slit Interference"

by Jess H. Brewer on 2005-03-22:

Huygens' Principle

[paraphrased] "Every point on an advancing wave front may be considered a source of outgoing spherical waves." The concept of a "wave front" is a little vague, of course; you can think of it as the "crests" of waves if you are visualizing waves in water or on stretched strings, but in 3 dimensional waves a "crest" corresponds to a locus of maximum (positive) amplitude of the wave. In general any locus of fixed phase will do just as well, as long as you use the same fixed phase (plus 2) to defined the adjacent "wave front". Naturally we make no attempt to draw 3D spherical waves on a flat page; all the 2D pictures are meant only as "conceptual shorthand". This will be even more abstract as we start drawing "phasor" diagrams.

Be sure to review your trigonometry - we'll be using it!

Interference from TWO SLITS

(Young's experiment modernized)

The "near field" intensity pattern (where "rays" from the two sources, meeting at a common point, are not even approximately parallel) is difficult to calculate, though it is easy enough to describe how the calculation could be done. We will stay away from this region - far away, so that all the interfering rays may be considered parallel. Then it gets easy!

Simplified sketch assuming incident waves hitting the barrier in phase (i.e. normal incidence) shows an obvious path length difference of = d sin between the waves heading out from the two slits at that angle. If this path length difference is an integer multiple of the wavelength we get constructive interference. This defines the nth Principal Maximum (PM):

d sin n = n

Often we are looking at the position of interference maxima on a distant screen and we want to describe the position x of the nth PM on the screen rather than the angle n from the normal direction. We always define x = 0 to be the position of the central maximum (CM) - i.e. = 0. If the distance L from the slits to the screen is >> d (the distance between the slits), as it almost always is, then we can use the small angle approximations sin tan so that n n /d and xn = L tan n L n giving xn n L /d.

Be sure you can do calculations like these yourself. Such problems are almost always on the final exam.

Time permitting, I will start on Multiple Slit Interference. The handout covers this in detail; if I don't cover it today, be sure to study the handout over the weekend!


"Multiple Slit Interference"

by Jess H. Brewer on 2005-03-26:

PHASORS ON STUN!

We now take you to a world beyond time and space, a world of pure mathematics where what you see are wave amplitudes and phases of different rays of a coherent wave with a given frequency and wavelength, interfering to make a combined amplitude - the world of The Phasor Zone. (Dew-dew-dew-dew, dew-dew-dew-dew, DEW-diddly-ew-dew...)

In this abstract world each wave is seen as an amplitude Ai pointing away from some origin at a phase angle i in "phase space" - a phasor. All the phasors representing different wave amplitudes are "precessing" about the origin at a common angular frequency (the actual frequency of the waves) but their phase differences do not change with time. Thus we can pick one wave arbitrarily to have zero phase and "freeze frame" to show the angular orientations (and lengths) of all the others relative to it.

Phasors are vectors (albeit in a weird space) and so if they are to be added linearly we can construct a diagram for the resultant by drawing all the amplitudes "tip-to-tail" as for any vector addition. If there are any configurations that "close the polygon" (i.e. bring the tip of the last phasor right back to the tail of the first) then the net amplitude is zero and we have perfect destructive interference!

For an idealized case of N equal-amplitude waves out of phase with their neighbours by an angle we will get a minimum when N = n(2), satisfying the above criterion. This is the condition for the nth minumum of the N-slit interference pattern; we usually only care about the first such minimum, which occurs where N = 2.

To see where in real space that first minimum occurs, we have to go back to the origin of the phase differences due to path length differences: /2 = / = d sin /, giving

d sin first min. = /N .

Note that this looks a lot like the formula for principal maxima, but it describes the angular location of the first minimum. This offers a good object lesson: Never confuse a formula with its meaning! You may memorize all the formulae you like, but if you try to apply them without understanding their meanings, you are lost. Note also that the central maximum is narrower by a factor of N than the angular distance between principal maxima. This is why we build "diffraction gratings" with very large N....



"Maxwell's Equations" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Maxwell's Equations"

by Jess H. Brewer on 2005-03-13:

Ampère's Law revisited

What happens when you run a current "through" an uncharged capacitor? Apply Ampère's Law around the wire; now extend the "open surface bounded by the close loop" so that it passes through the gap in the capacitor without cutting any current-carrying wires. (Imagine that you are making a big soap bubble with a hoop.) Does the magnetic field around the loop suddenly disappear? I think not!

Maxwell proposed a "time-varying electric flux" term symmetric to the changing magnetic flux in Faraday's Law to resolve this paradox. Suddenly a time-varing electric field generates a magnetic field, as well as the reverse.

So now we have Gauss' Law in two forms (integral over a closed surface vs. differential at any point in space) for E (or, better yet, for D = E) and for B (where it may seem trivial to express the fact that there don't seem to be any magnetic "charges" [monopoles] but in fact this is quite useful).

We have Faraday's Law also in two forms; we will only be using the integral form in this course, but you should be able to recognize the differential form.

And we have Maxwell's corrected version of Ampère's Law which again we will be using here only in the integral form but you should be able to recognize in either form.

These 4 Laws constitute Maxwell's Equations, which changed the world. To complete "everything you need to know about electromagnetism on one page" you should include the Lorentz Force Law (including the electric force) and the Equation of Continuity (which simply expresses the conservation of charge). That's it. Real simple "cheat sheet", eh?

From Ampère's Law applied to a specific geometry we have the first mixed time- and space-derivative equation. I will derive this today and then move on to the next equation which comes from Faraday's Law.



"The Magnetic Field" Topic

Found 3 Lectures on Mon 07 Oct 2024.

"I x B: the Lorentz Force"

by Jess H. Brewer on 2005-02-16:

UNITS

The Coulomb is defined as an Ampere-second, and an Ampere is defined as the current which, when flowing down each of two parallel wires exactly 1 m apart, produces a force per unit length of 2x10-7N/m between them. No kidding, that's the official definition. I'm not making this up!

To get the definition of a Tesla [T] we have to wait until the next Chapter on where magnetic fields come from, i.e. the Law of Biot & Savart.

Circulating Charges

If v is perpendicular to B, we get a very familiar situation: the force on the particle is always normal to its velocity, so it cannot change its speed; and yet it is constantly accelerated. Ring a bell? Come on, you know this: it's good 'ol uniform circular motion! Solve the familiar equation (v2/r = QvB/m and p = mv) to get

p = Q B r

where p is the momentum and r is the radius of the orbit. "It turns out" that this relation is relativistically correct, but you needn't concern yourself with this now. Since v = r , this means = QB/m, a constant angular frequency (and therefore a constant orbital period) regardless of v! (This, unfortunately, is not relativistically correct.) Faster particles move in proportionally larger circles so that the time for a full orbit stays the same (as long as v << c). This is what makes cyclotrons possible. At TRIUMF, since v ~ c, we have to resort to an ingenious trick to compensate for relativity.

Wien Filters

If v, B and E are all mutually perpendicular, the particle will pass undeflected iff E = vB. This makes a nice velocity selector. If you also measure the radius of curvature of the same particle's path in B with no E, you know its momentum. Putting these together gives you the ratio Q/m. If you know Q (which was not so easy until Milliken's "oil drop" experiment) then you know m. This is the basis for conventional mass spectroscopy. However, the cyclotron is an even better mass spectrometer. Why?


"What B Do & Do B, Do!"

by Jess H. Brewer on 2005-02-23:

Biot & Savart vs. Ampère

now that you know how to do integrals, you're expected to use them! (Dang! Ignorance is easier!)

Law of Biot & Savart

Circular Current Loop via Biot & Savart: too hard to calculate the field anywhere except on the axis of the loop. There (by symmetry) the field can only point along the axis, in a direction given by the RHR: curl the fingers of your right hand around the loop in the direction the current flows, and your thumb will point in the direction of the resulting magnetic field. (Sort of like the loops of B around a line of I, except here B and I have traded places.) As usual, symmetry plays the crucial role: current elements on opposite sides of the loop cancel out each other's transverse field components, but the parallel (to the axis) components all add together. As for the electrostatic field due to a ring of charge, we get the same contribution to this non-canceling axial field from each element of the ring.


"Link the Loop with Symmetry: Ampère's Law"

by Jess H. Brewer on 2005-02-26:

Ampère's Law

The integral of B// dl around a closed loop (where B// is the component of B along the path at each element dl) is equal to µ0 times the net current Iencl linking the loop (i.e. passing through it).

(Used like Gauss' Law only with a path integral.)

Long Straight Wire via Ampère's Law: It's so easy!

Any Cylindrically Symmetric current distribution gives the same result outside the conductor; inside we get an increase of B with distance from the centre, reminiscent of Gauss' Law....

Circular Current Loop via Ampère's Law: Forget it! Ampère's Law is of no use unless you can find a path around which B is constant and parallel to the path. There is no such path here.

Torque on a Current Loop & the Magnetic Dipole Moment

It can be shown in detail that the torque on a rectangular loop of area A carrying a current I in a magnetic field B is given by the vector ("cross") product of µ with B, where µ = I A n and n is the unit vector normal to the plane of the loop (taken in the sense of the RHR for the current around the loop). Stated without proof (SWOP): the shape of the loop doesn't matter.

By the same logic as for electric dipole moments in electric fields, the potential energy of the magnetic dipole in the magnetic field is minus the scalar ("dot") product of µ with B. This may be familiar from Thermal Physics.

For details see PDF or printer-friendly gzipped PostScript files.



"Thermal Physics" Topic

Found 6 Lectures on Mon 07 Oct 2024.

"First Class!"

by Jess H. Brewer on 2005-01-06:

I will be making an unconventional introduction to Thermal Physics, based on the microscopic approach of Statistical Mechanics which is usually withheld until later years, for reasons with which I disagree. You'll thank me later.

Much of the lecture was presented using Open Office, a free, Open Source replacement for Micro$oft Office. You are welcome to download the PDF file if you like; I will make the Open Office or PPT file available on request.


"Temperature"

by Jess H. Brewer on 2005-01-07:

Put two systems in thermal contact (meaning that they can exchange energy freely). What tends to happen is whatever increases the total number of possibilities - i.e. the multiplicity and therefore the entropy of the combined system.

We have to remember that the total energy U is conserved. Thus dU1 = - dU2.

A maximum of the total entropy occurs where its rate of change with respect to U1 is zero.

Working this out in detail gives a definition of temperature.

We use this definition to examine the thermal behaviour of an unusual system: N spin 1/2 electrons in an applied magnetic field B. The exotic features of this system are due to its unusual feature of having a limit to the amount of energy U it can "hold". We correctly expect that the number of ways it can have that maximum energy (where all the spins are "up") is 1, so at the maximum U the entropy is zero; since it is nonzero at lower U, it must be decreasing with U for energies approaching the maximum. Thus the slope of entropy vs. energy starts positive, goes down through zero and then becomes negative. Since this is the inverse temperature, the temperature itself starts low, goes to infinity, flips to negative infinity and finally approaches zero from the negative side. What does this mean?!

Negative temperatures exist. It is easy to make them in the lab. They are hotter than positive temperatures (even hotter than infinite positive temperature!); the hottest temperature of all is "approaching zero from below". This weirdness is the result of our insistence that "hot" must mean "high temperature", requiring the definition of temperature as the inverse of the slope of entropy vs. energy. Live with it.

UNITS

Another silly convention we have to live with is the idea that temperature should have its own special units, "degrees" Kelvin or K. This is absurd. One look at the definition of temperature tells us that it is measured in energy units. Thus "K" is an energy unit: 1 K is equal to 1.3806505 x 10-23 J. OK, it's a very small energy unit, but the only reason it is not some nice round number (like 10-23 J) is that it was made up arbitrarily before anyone knew what temperature really was. Sorry. You'll just have to cope with it. By the way, that conversion factor is known as Boltzmann's constant, kB = 1.3806505 x 10-23 J/K. When you talk to an engineer about entropy, you had better express it in units of kB rather than as a pure number as I have defined it.


"The Boltzmann Distribution"

by Jess H. Brewer on 2005-01-10:

Big Reservoir and Little System

Suppose the two systems in thermal equilibrium are a huge, complex heat reservoir R and a tiny, incredibly simple system S that is so minute and trivial that we can realistically talk about which fully specified microstate "" it is in. We are allowed (indeed, we are encouraged) to narrow our focus to just one degree of freedom of one particle (or whatever), such as the "system" consisting of the orientation of the spin of a single electron.

The energy contained in this small system S in state "" is called . We imagine that this energy was removed from the reservoir R, to make its energy UR = U - , where U is the total energy of the combined systems (and was the energy of R before we tapped some off into S).

This process changes the entropy of R by an amount . . . well, this is more easily displayed in a PDF file or (if you want a more printer-friendly format) a gzipped PostScript file.


"Ideal Gases: Energy and Pressure"

by Jess H. Brewer on 2005-01-13:

EQUIPARTITION OF ENERGY:

PRESSURE: A single particle bouncing around in a box with perfectly elastic specular collisions causes an average force on the walls of the box.

IDEAL GAS:

As usual, a more complete graphical summary is available in PDF or gzipped PostScript format.


"Particle in a Box"

by Jess H. Brewer on 2005-01-13:

Discussion of standing waves, quantization and de Broglie's Principle:

= h/p

[For an introduction to Quantum Mechanics in the form of the script to a comical play, see The Dreams Stuff is Made Of (Science 1, 2000).]
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.

.

Discrete wavelengths, momenta and energies. Lowest possible energy is not zero. As the box gets smaller, the energy goes up!

Handwaving reference to black holes, relativistic kinematics, mass-energy equivalence and how the energy of confinement can get big enough to make a black hole out of even a photon if it is confined to a small enough region (Planck length).

For more details see PDF or print-friendly gzipped PostScript files.


"Momentum Space"

by Jess H. Brewer on 2005-01-15:

Allowed states are evenly spaced in momentum but not in energy, which is what we want in our Boltzmann distribution. Since p E, we expect (E) 1 / E. (Sketch.)

Moving to 3-D picture, there is one allowed state (mode) per unit "volume" in p-space. But if what we want is the density of states per unit magnitude of the (vector) momentum, there is a spherical shell of "radius" p and thickness dp containing a uniform "density" of allowed momenta whose magnitudes are within dp of p. This shell has a "volume" proportional to p2 and so the density of allowed states per unit magnitude of p increases as p2. This changes everything!

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.

.

The details are on the Momentum Space handout and in the PDF and printer-friendly gzipped PostScript files from the graphical presentation in class.

You may feel this is going too far for a First Year course, and I have considerable sympathy for that point of view. I simply wanted you to have some idea why the Maxwellian energy and speed distributions have those "extra" factors of E and v2 in them (in addition to the Boltzmann factor itself, which makes perfect sense). The textbook (perhaps wisely) simply gives the result, which is too Aristotelian for us, right?

Rest assured that I will not ask you to reproduce any of these manipulations on any exam. At most, I will ask a short question to test whether you understand that one must account not only for the probability of a given state being occupied in thermal equilibrium (the Boltzmann factor) but also how many such states there are per unit momentum or energy (the density of states) when you want to find a distribution.



"Waves" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Wave Review"

by Jess H. Brewer on 2005-03-15:

Today I want to spend some time reviewing the general properties and behaviour of waves. Only a few of the topics will be new, but for the rest of the course I am going to be relying on your deep and intuitive understanding of how waves behave, so I will not just rely on what you learned last term.

Simple Harmonic Motion (SHM) in Time and Space

. . . a review of sinusoidal travelling waves.

Solutions of the Wave Equation

. . . the linear Wave Equation has solutions that are not sinusoidal. In fact, any well-behaved function of only u = x - ct, where c is the wave's propagation velocity, will automatically satisfy the Wave Equation.


"Wave Review cont'd"

by Jess H. Brewer on 2005-03-18:

Today I will continue reviewing the general properties and behaviour of waves. See also the condensed version (displayed in class) as a PDF file.

The Electromagnetic Spectrum

. . . from ~1 Hz seismic waves (wavelength ~108 m) to ~1020 Hz gamma rays (wavelength ~10-12 m). We will, out of human biological chauvinism, pay most attention to the visible spectrum between ~400 and ~800 nm in wavelength.

Actual Wave Functions: Plane and Spherical Waves

The standard "plane wave" propagating in the z direction can be generalized to propagate in the k direction, where k is called the wave vector. It has the same magnitude as usual, k = 2/, but the scalar kz is replaced by the dot product kr (where r is the vector position where we want to know the wave's amplitude). Imagine the wave "crests" as plane sheets stretching off to infinity in both directions perpendicular to k, marching along in the k direction at c. Obviously the plane wave is an idealization. We won't use this formulation explicitly very often, but it serves to remind us that the wave has a well defined direction of propagation, which we habitually express in the form of rays, a picture inherited from Newton, who insisted that light was particles following trajectories like little billiard balls, until Huygens showed that it was indeed waves. (We now know they were both right!)



"Weird Science" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Weird Science: E&M"

by Jess H. Brewer on 2005-01-18:

Start with a brief comment on "weirdness" in Physics.

Continue with a brief review of 3-dimensional vectors. Make sure you can do all the operations "in your sleep", both analytically (using algebra and the left hemisphere of your brain, which is reputed to handle abstract symbolic logic) and graphically (using various physical analogues and the right hemisphere of your brain, which is said to govern intuition and spatial vision). Whatever tricks you use to remember the "right hand rule" convention for "cross products", be sure they are well practiced; we'll be using them a lot when we get to Magnetism!

Then on to our first topic in Electricity & Magnetism (E&M): the Coulomb force between electric charges.

A comparison of the gravitational force between masses with the electrostatic force between charges shows just two differences:

  1. There are no negative masses; gravity is always attractive, whereas there are both positive and negative charges, so that the electrostatic force can be either attractive (for unlike charges) or repulsive (for like charges).
  2. The electrostatic repulsion between two electrons (for example) is about 1043 times stronger than their mutual gravitational attraction. That's a lot!
Nevertheless, these two "force laws" are handled in exactly the same way and everything you learned about gravity is (at least mathematically) applicable to Classical Electrostatics. Some discussion ensues about just how much we really understand about Gravity. I will ignore General Relativity (because I don't understand it!) and assume we aready know all about Gravity, from first term. Be sure it's true!

As usual, details are available in either PDF or printer-friendly gzipped PostScript format.




"What Does It All Mean?" Course

"Elementary Particles" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Small Stuff"

by Jess H. Brewer on 2005-06-17:



"Introduction" Topic

Found 2 Lectures on Mon 07 Oct 2024.

"Introduction"

by Jess H. Brewer on 2005-06-10:

The title and brief description above have enticed you to sign up for this course, but I really have no idea what you expect or desire from me. An instructor's usual response to such circumstances is to proceed according to the syllabus or the prerequisites for following courses (difficult in this case, as there is neither a syllabus nor any course to follow) or according to whim ("This is what I like to talk about; if they don't like it, tough!") but after 28 years of delivering lectures I've had my fill of playing Expert/Authority figure, and presumably you have no need for me in such a role. So we're going to do it differently.

First I need to know a bit about you and your expectations/preferences. Who are you? How much do you already know about Physics? How seriously do you take Poetry? What did you think this course was going to be about? What would you like this course to be about? Do you expect to do any homework? Reading? Do you mind doing some things on the computer? Do you have access to the Web? (If the consensus is negative on the last question, then you are probably not reading this; so you can tell I am hoping to be able to use Web tools with the course.)

While I am a tireless advocate for Poetry, I have no credentials as a Poet, and there are bound to be at least some of you who do; so I will never be tempted to speak with Authority about that discipline - all my pronunciamentos will be understood to represent only my own opinion, and counteropinions will be welcome. Just don't go all ad hominum on me, OK?

I do have some Physics credentials, however undeserved, and I have a few favourite topics I'd love to weave into this short week if I can. I'll list a few of them below and ask you to give me some feedback on which you'd like me to concentrate upon.

There's more, of course, but we'll build on your preferences and follow the discussion where it leads.


"Emergence"

by Jess H. Brewer on 2005-06-13:

Tacit Knowledge

Before we can "grow new language" we need to have a vocabulary of familiar "old" words to juxtapose in unfamiliar ways. In Physics everything starts from Classical (Newtonian) Mechanics, which in turn starts from the familiar equation F = m a, where F is the net force exerted on some body, m is its mass and a is the resulting acceleration. This isn't actually the way Newton expressed his "First Law", but it will do. Most people are fairly familiar with this Law by the time they reach University, so it serves as an example of what Michael Polanyi would call "Tacit Knowledge" - things we know so well they are "obvious" and/or "Common Sense".

Emergence

In the The Skeptic's Guide chapter on Mechanics I show how F = m a can be "morphed" by mathematical identities into principles that appear to be different, like Conservation of Impulse and Momentum, Conservation of Work and Energy or Conservation of Torque and Angular Momentum. There is really nothing new in these principles, but we gain insight into the qualitative behaviour of Mechanics from the exercise. Thus new Common Sense emerges from the original language by a process analogous to metaphor in Poetry.

In the same way, bizarre phenomena like superconductivity emerge in the behaviour of many crystals, even though every detail of the interactions between their components is completely understood. When the familiar is combined in new ways, the unfamiliar emerges, and it is often very unfamiliar. This seems to be characteristic not only of what Physicists do, but also of how Nature behaves!

Starting Points

The paradigms of Newtonian Mechanics are only part of the vocabulary we need to begin constructing the metaphors of Relativity and Quantum Mechanics. We also require a basic understanding of the attributes of Waves, like frequency, wavelength and amplitude. This will keep us busy today.



"Quantum Mechanics" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"Particle in a Box"

by Jess H. Brewer on 2005-06-17:

In 1924, Prince Louis Victor Pierre Raymond duc de Broglie hypothesized in his 25-page dissertation that all particles are also waves, and vice versa, with their momentum p and their wavelength related by

= h/p     and     p = h/

This later won him a Nobel Prize. Nice thesis!

Whatever we might "mean" by this, it has some dramatic consequences: imagine that we have confined a single particle to a one-dimensional "box". Examples would be a bead on a frictionless wire, or an electron confined to a long carbon nanotube (or a DNA molecule); both of the latter two examples are currently being studied very enthusiastically as candidates for nanotechnology components, so they are not the usual frivolous Physics idealizations!

If the particle is really like a wave, then the wave must have nodes at the ends of the box, just like the standing waves on a guitar string or the sound waves on a closed organ pipe. This means there are discrete "allowed modes" with integer multiples of /2 fitting into the length L of the "box". Not all wavelengths are allowed in the box, only those satisfying this criterion; therefore, not all momenta are allowed for the particle bouncing back and forth between the ends of the box, only those corresponding to the discrete ("quantized") allowed wavelengths.

Since the kinetic energy of the particle increases as its momentum increases, the lowest allowed energy state is the one whose wavelength is twice the length of the box, and if the box shrinks, this "ground state" energy increases. Moreover, since the particle is bouncing back and forth off the ends of the box (like a ping pong ball between the table top and a descending paddle), the average force exerted by the particle on the walls of its confinement increases as the walls close in.

Is this not a lovely metaphor? Like most people, every particle (because it is also a wave) cries, "Don't fence me in!" and will resist confinement with ever increasing vigour as the walls close in.

This resistance eventually goes beyond mere force. If you will allow me to state without adequate explanation that Einstein's famous equation "E = m c2" means not only that any mass m represents a large amount of energy E, but also that energy stored up in a small region has an effective mass, with all the concomitant effects such as gravitational attraction for other masses, then you will see that as the confined particle's energy increases (due to tighter and tighter confinement) it begins to have a gravitational field. And if its energy increases enough it will act as a "black hole" for other objects within L of the box - including the walls of the box! At this length scale (called the Planck length) all bets are off - we do not understand physics at this level of "quantum gravity", although armies of Physicists are now working on it.

So the humblest particle, even the photon (which has no rest mass), will eventually dismantle its jail even if it has to deconstruct the very Laws of Physics to do so. A fine example for us all, I think, and an apt mascot for Amnesty International!



"Waves" Topic

Found 1 Lectures on Mon 07 Oct 2024.

"This, That & Waves"

by Jess H. Brewer on 2005-06-17:

We spent quite a bit of time today in unrehearsed discussion on spontaneous topics like radiation hazards, cancer therapies and other stuff. I count such days among my favourites, but sometimes people who have paid good money to hear about specific topics feel shortchanged by such free discussions. I hope we can strike a balance that is satisfactory to all; if not, well, you can't please everyone, so you have to please yourself. :-)

Towards the end I did manage to get started talking about the implications of de Broglie's hypothesis that all particles are also waves, and vice versa, with their momentum p and their wavelength related by

= h/p     and     p = h/

but the denouement had to wait for tomorrow.




"UBC Physics 401" Course

"Conservation Laws" Topic

Found 4 Lectures on Mon 07 Oct 2024.

"Poynting Away!"

by Jess H. Brewer on 2006-01-13: