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Lectures in PHYS-4400 (1)

Updated: 03 May 2011 Tuesday.


n      10    10
hi    165    200
lo     75    133
ave   135.5  176.3
s.d.  28.76  21.23


Week of April 25-29, 2011.

Monday 4/25: Office hours.

Tuesday 4/26: FINAL EXAM (2 HOURS) 8-10am. Office hours.

Wednesday 4/27: (PHYS-2050 FINAL EXAM (2 HOURS) 2:45-4:45pm). Office hours.

Thursday 4/28: NOTE: No office hours.

Friday 4/29: LATE FINAL EXAM (2 HOURS) 11am-1pm.Office hours.

Week of May 2-6, 2011.

Monday 5/2: Office hours.

Tuesday 5/3: Grades will be done by Noon.

Week of January 10-14, 2011.

Monday 1/10: Class begins. We begin by locating this course in the overall study of Physics. Next we recognize that it has been some time for most people since they had PHYS-2070 (or equivalent) Introductory E&M, so we start this week by doing a quick review of basic E&M: The simple hydrogen atom -- whatever charge is, the charge on the electron (-e) and the proton (+e) exactly cancel. The Electric Force between two point charges, Coulomb's Law looks like Newton's Law of Universal Gravity. Real Electric Charges. Two charges: like charges repel, unlike (opposite) charges attract. 1 Coulomb of charge is an enormous amount of charge. Two 1.00 C charges separated by 1.00 meters have a force of nine-billion Newtons acting on each other. Four Fundamental Forces in Nature: Gravity, E & M, Weak Nuclear Force, Strong Nuclear Force. The Hydrogen Atom: Gravity loses to Electric Force by a factor of 200 million dectillion (!!!). The Helium Atom: Putting more than one proton in the nucleus produces enormous forces on the tiny protons -- Need the Neutron and the Strong Nuclear Force (!!!).

Tuesday 1/11: Finding the net vector electric force FE for a system of point charges. Remember: In PHYS-2070, Looking at Symmetry and Zeroes (problems where the answer is zero) as a way of solving problems. How does q1 know that q2 is there? -- "Action at a Distance" -- Gravity and the Electric Force are not contact forces. The mathematical construct of the Electric Field. E is not an observable quantity. (Side example: Methods of measuring speed v, do not directly measure speed v.) Electric Field is a vector. FE = q E. For a point charge, E = k q1 /r2. SI units for E-field: (N/C). E-field lines radiate away from a positive point charge; converge towards a negative point charge. If the universe is charge neutral, can have all E-field lines from + charges terminating on - charges.Why use E-fields, when you need the force F = q E anyway? Because it allows us to examine the environment without needing another charge. Direct integration of Electric Force and Electric Field are similar, so we'll just go over direct integration of the E-field. Charge distributions -- lamda (linear charge density, C/m), sigma (surface charge density, C/m²), rho (volume charge density, C/m³). Note the similarity to mass distributions from PHYS-2050. Examples: Rod in-line with line from point P (1-dimensional integration). Rod perpendicular to line from point P. Note that in all these cases, we can predict the long range behavior (E-field behaves as a single point net charge), and anticipate the close-in short range behavior. Electric Flux: Electric field times Area. Analogy of a bag or box around a light, captures all the light rays no matter the size or shape. Use known E-field of a point charge to evaluate what the Electric Flux must be equal to. Review of Dot Product. Gauss' Law for Electricity. Using Gauss' Law for Point Charge, Conducting Sphere (case 1: r < R). Note that E-field is zero inside a spherical conducting sphere (solid or hollow). If the Earth were hollow, there'd be no gravity inside the Earth either, besides being zero-gee at center of core. Using Gauss' Law for Point Charge, Conducting Sphere, Insulating Sphere, Infinite Line of Charge. P.E. is minus the Work. Potential V is similar, but the integral is done on E-field not Force. More importantly the Potential V is an observable quantity. Find components of E by negative of the partial derivative of Electric Potential function V. It will turn out that charge accumulates on the tips of long pointy things -- applies in why some things seem to always get hit by lightning (golfers, people standing in an open field, church steeples). Emax = 3,000,000 N/C = 3,000,000 V/m, in dry air. Ben Franklin and lightning rods. Why your hair stands up warning you that you are getting charged. Handy chart of the four quantities: FE (vector, 2 charges), E (vector, 1 charge), UE (scalar, 2 charges), V (scalar, 1 charge) .Simplified equation V = E d. (But remember that it's really delta-V = - E d .) Example: Lightning.

Thursday 1/13: Distribute Syllabus. Moving from Field Theory to Applications leading to Devices. Start of Capacitors and Capacitance. The Capacitor stores charge +Q on one plate and -Q on second plate, stores energy in the E-field between the plates. This is different from a battery, which has energy stored in its chemical reaction. Capacitor Equation. SI unit for Capacitance is the Farad. 1F is a large capacitor. Usually deal with µF (microfarad = 1/1,000,000th of a Farad) and pF (picofarad = 1/1,000,000,000,000th of a Farad). Apply Gauss' Law for Electricity to the constant E-field of the Parallel Plate Capacitor. We now have an "operational equation", true for all capacitors, and a "by geometry" equation for the special case of the parallel plate capacitor. Work to assemble charges on a capacitor = Energy stored in the capacitor = U = ½CV² . Making a real capacitor. What if not filled with air? Filling with conductor, must have at least one gap, otherwise will short outthe plates. A conducting slab inside a parallel plate capacitor makes two capacitors in series. Charge neutral slab stays charge neutral, but +Q of top plate attracts -Q on top of slab, and -Q of bottom plate attracts +Q on bottom of slab. Dielectrics -- an insulator where the +/- charge pairs are free to rotate, even if they do not move. Dielectric constant (kappa) and Dielectric strength (E-max). (See Table 26-1, p. 736) Dielectic constant increases capacitance over air gap. Dielectric strength usually bigger than Emax in air. Both allow you to (a) make bigger capacitors (or smaller for the same values) and (b) make non-hollow, self-supporting components. Electrolytic capacitors -- must be connected into the circuit with correct + and - polarity. Resistance vs. Conductance. Ohm's Law: V=IR form. (Ohm's "3 Laws") We usually treat the wires in a circuit as having R=0, but they usually are not superconductors. Resistance is a function of temperature. Kammerleigh Onnes 1916 work on extending the R vs. T curve toward T = 0 Kelvin. Discovered Superconductivity, where R=0 identically. Resistance by geometry. R = rho (L / A), where rho = resistivity of the material, L = length and A = cross-sectional area. Magnetic Force on a Moving Electric Charge - The Cross Product and Right-Hand Rule (R.H.R.). The Cross Product (or Vector Product) is the exact opposite of the Dot Product (or Scalar Product). Multiplying two vectors together by a cross product gives us another vector (instead of a scalar). And the cross product is not commutative, vector-A × vector-B = - (vector-B × vector-A), so the order is paramount. Using Right Hand Rule to assign directions to x,y,z coordinates. Constant speed, perpendicular constant magnetic force --> Uniform Circular Motion. Cyclotron frequency -- no dependence on the radius (constant angular velocity). Velocity Selector - the Magnetic Force is speed dependent, the Electric Force is not. So we can use an E-field to create an Electric Force to cancel the Magnetic Force on a moving charged particle, such that at the speed v = E / B, the particle travels exactly straight with no net force -- any other speed and the particle is deflected into a barrier. Hence a velocity selector "selects" velocities... A current carrying wire consists of moving electric charges, and so therefore would see a magnetic force from a magnetic field. Discussion of microscopic theory of charges in a conductor. Drift velocity is the very slow net movement of the electrons moving randomly in the wire. Magnetic Force on a Current Carrying Wire. Demo -- hey it works and even in the right direction! Technically current is not a vector, despite the fact we talk of direction of current. J = current density = current/cross-sectional area is the vector related to current. NOTE: J-vector = sigma × E-vector (current density = conductivity × E-field) is the vector version of Ohm's Law.

Friday 1/14: Q1 and your PID number. (If you missed Quiz 1 you will be able to get some of the points by downloading Quiz 1A from the website and turning it in.) The Biot-Savart Law. B-field from a infinitely long straight current carrying wire by direct integration. Gauss' Law for Magnetism. Not as useful as Gauss' Law for Electricity, because it is always zero (no magnetic monopoles). However, there is something we can use in a similar way which involves involving a path integral along a B-field and the current(s) contained inside -- Ampere's Law. Use in a way similar to the way we used Gauss' Law for Electricity. Use symmetry and geometry to select your Amperean Loop to your advantage. 3-D directions and R.H.R. You can make a list of axes directions or unit vectors (x y z x y z) and (i j k i j k) and find the 3rd direction of the cross product by going to the right (+) or to the left (-) in the list. Example: i-hat × j-hat = +k-hat, since order is "i j k", but j-hat × i-hat = -k-hat, since "j i k" goes to the left. Ampere's Law. Use in a way similar to the way we used Gauss' Law for Electricity. Use symmetry and geometry to select your Amperean Loop to your advantage. Faraday's Law of Induction. A changing magnetic flux induces a current, induces an e.m.f., in the circuit, substituting for the battery as the power source. Lenz's Law "of maintaining the status quo." The coil acts as if it opposes any change of the magnetic flux inside, by inducing a magnetic field to cancel and increasing flux or maintain a decreasing flux. To create this induced magnetic field, one needs an induced current, which is powered by an induced emf. It is Lenz's Law that gives us the minus sign in Faraday's Law of Induction. -- it doesn't work properly in the gap between the plates of a capacitor while it is charging. So James Clerk Maxwell fixed it with a "displacement current" term, involving the time derivitive of the Electric flux in the gap. Ampere-Maxwell Law. Maxwell's Equations in integral form. Note that Maxwell didn't invent the four equations, only half of one, but he figured out what todo with them. E & M Waves. In vacuum (free space): a traveling set of perpendicular E-fields and B-fields, as sine waves constantly changing in space and time, moving with wave speed c (the speed of light in vacuum). Review of the Del operator. Partial derivatives, gradient, divergence, curl, Del-squared.

Week of January 17-21, 2011.

Monday 1/17: MLK Day -- No Classes.

Tuesday 1/18: Working with 3rd and 4th of Maxwell's Equations to generate partial differential equations of E(x,t) and B(x,t). (see pp. 958-959 in Serway) Looking at the solution to Traveling E-M Wave, with v in x-direction, E in y-direction and B in z-direction. Angular frequency omega, wave number k. c = Emax / Bmax.

Thursday 1/20: Modern Physics -- goes to size/time/length scales far outside our normal experience. Classical Relativity (two observers, two frames of reference), Special Relativity (speed constant), General Relativity (accelerations or gravity). Einstein's postulates: (1) All observers see the same Physics laws. (2) All observers measure the speed of light in vacuum as c. Beta, gamma, Length Contraction and Time Dilation. Alpha Centauri is 4.20 LY from Earth (proper length). Those on a starship see a different distance and experience a different time than the observer left on the Earth. But both think the other observer is moving at v < c. No preferred observer in Special Relativity. Two observers cannot agree on what they see, distance or time. They can only agree that the speed of light in vacuum is c. One sees the proper length: a length measurement where both ends are measured at the same time. One sees the improper length: a length measurement made at two different times. Neither observer is preferred -- that is one is not "more right" than the other. They are both right. These differences in time and length measurements have been confirmed by experiment. Experimental confirmation of Special Relativity: put atomic clocks on aircraft, spacecraft. Two observers cannot agree on the order of events, either. The concept of "simultaneity" is gone. Another confirmation of Special Relativity: Muons (a form of heavy electron) are created in the upper atmosphere -- they're unstable and will decay. Muons measured at mountaintop -- by sea level, nearly all should have already decayed. But you detect almost as many at sea level as on the mountaintop, because the muon lifetime is measured in the muon's rest frame not while we are watching it moving. The Correspondence Principle -- at some point our Classical Physics results need to match the Modern Physics results. So when do we need Special Relativity? For eyeball measurements, we have trouble distiguishing the size of things that are only off by 10%. That would correspond to a gamma = 1.10, and a beta = 0.417 c. Difference in time with identical clocks left on the ground. Quiz 2 Take-Home handed out Thursday 20 January 2011 on reviewing the Del operator, due Tuesday 25 January 2011.

Friday 1/21: Apologies for cancelling class -- it normally takes me 1 hour 15 minutes to drive in. Friday it took over 2 hours -- there was no way, given how fast cars were traveling on the slick roads, that I could get to Kalamazoo in time. Sorry for anyone inconvenienced.

Week of January 24-28, 2011.

Monday 1/24: Prologue (Chapter 0): Brau's review of E&M uses more technical versions of the equations we used in PHYS-2070. For Coulomb's Law and the Electric Field, it looks like we now have an inverse cube law ( 1/r³ ) instead of inverse square ( 1/r² ). But this is an illusion, brought on because we are using r-vectors in the top of the fraction and not r-hat unit vectors. Previously we saw Maxwell's Equations in integral form. Now we have Maxwell's Equations in differential form.

Tuesday 1/25: Chapter 1: Previous waves involved some sort of medium -- vibrating strings, vibrating drumheads, sound in air, waves in water, etc. A disturbance traveled through the material, the material itself only undergoes small displacements. If Maxwell was right and light is an E-M wave, then what is waving in free space (vacuum)? It was postulated that there had to be an "aether" -- spreads throughout space, no mass, can't be seen, but has extremely high tension to account for visible light vibrations. Fizeau's experiment for determining speed of light in flowing water. At first it was thought to confirm the possibility of an aether, but others showed it was possible to come up with the same terms without an aether. Michelson-Morley experiment showed no variation in speed of light due to the background flow of any aether. The speed of light in vacuum is the same, regardless of direction or motion. Therefore there cannot be an aether. This leads eventually to Einstein's postulates for relativity. Previously we looked at 1-dimension plus time. Now we want to look at 3 spatial dimensions (x, y, z) and 1 temporal dimension (ct) -- by looking at ct and not t, this fourth dimension has the same units as the others. Minkowski space and the world line of an event. Light cone -- event at origin, information about the event (past and future) must lie within the line cone. For Euclidean geometry in 3-dimensions, the square of the magnitude of the differential displacement is always going to be positve. For the Pseudo Euclidean used for relativistic Minkowski space, we add minus signs to the spatial part, so that the square of the magnitude of the differential displacement will be ds² > 0 for time-like events, ds² = 0 for light-like events and ds² < 0 for space-like events. The ds² = 0 result will make sense when we realize that if you are a photon, a particle of light, there is no time to the universe and the universe has zero length.

Thursday 1/26:

Friday 1/27: Q2 Solution handed out. Topic 1 assigned.

  • Note: The actual Topic 1 Handout is 27 pages long. You've been given pages 1,2,11 and 27, and the link to the Topic 1 Assignment webpage. There you can see the whole handout as a PDF or as a Searchable HTML page with links to jump to the main topics.
  • If this all seems like Too Much Information, come see Dr. Phil and he'll help you find something of interest.
  • Please remember that you are advised NOT to choose a book you have already read.
  • Week of January 30-February 4, 2011.

    Monday 1/31:

    Tuesday 2/1:

    Thursday 2/3:

    Friday 2/4: Exam 1. (Re-Rescheduled)

    Week of February 7-11, 2011.

    Monday 2/7: Multi-pole moments. Dipoles, quadripoles.

    Tuesday 2/8: Showing that solutions for Phi in boundary values are the same solution. Work to assemble charges on multiple conductors in a capacitor. NOTE: If you are wondering why the Cij matrix is the inverse of the Kij matrix, recall that for a simple parallel plate capacitor, the energy stored is Uc = Q²/2C,

    Thursday 2/10: The parallel plate capacitor. Method of Images. Start of Separation of Variables. Quiz 3 Take-Home on the integral and differential forms of Gauss' Law for Electricity in Spherical Coordinates, due Tuesday 15 February 2011.

    Friday 2/11: Separation of Variables. PHI(x,y,z) = X(x) Y(y) Z(z) from Section 3.2.4.

    Week of February 14-18, 2011.

    Monday 2/14: Laplace's Equation in Spherical Coordinates. Separation of variables. The Legendre Polynomials -- The Rodriguez Equation.

    Tuesday 2/15: Solving for Legendre Polynomials. Spherical Harmonics.

    Thursday 2/17: Physical vs. "pure" multipoles. Showing that multipole expansion of an arbitrary charge distribution at large distance gives us the sum of Legendre polynomials, Pl(cos(theta)). Quiz 4 Take-Home on orthogonality of sines, due Tuesday 22 February 2011.

    Friday 2/18: If a charge distribution has a net charge Q, then for large values of r, the monopole term should dominate the potential V. If Q=0, then the dipole term should dominate for large r, unless the dipole moment p is zero. For physical dipoles, the location of the origin will affect the dipole moment.

    Week of February 21-25, 2011.

    Monday 2/21: Dr. Phil has canceled his classes due to treacherous roads.

    Tuesday 2/22: Return X1.

    Thursday 2/24: Exam 2.

    Friday 2/25: Spirit Day. (No Classes)

    Week of February 28-March 4, 2011.


    Week of March 7-11, 2011.

    Monday 3/7: Inducing a dipole moment in a neutral atom by an externally applied E-field. CRC Handbook of Chemistry & Physics data: makes sense that atoms in the periodic table with 1 s electrons (H, Li, Na, K, Cs) have much larger atomic polarizabilities (alpha) than filled shell noble gasses (He, Ne, Ar, etc.)

    Tuesday 3/8: Using atomic polarizability of a spherical shell of electrons to compare to Hydrogen result. (Using a quantum mechanical charge density rho(r) will be HW5 -- see below.) Some molecules have a permanent dipole moment, such as water. Effects of having E-field parallel or perpendicular to the line of the molecule. The general polarizability tensor in 3-D. Torques caused by dipole moments not aligned with applied E-fields.

    Thursday 3/10: Interpretting polarization as bound surface charges (sigma-sub-b) and bound volume charge densities (rho-sub-b).

    Friday 3/11: Return X2. The bound surface and volume charge densities are not "fictitious" charges, like the image charge method we used with conductors, but actually charge separations as a result of either induced or permanent polarization. Quiz 5 is actually it's Griffiths p. 170 Problem 4.11, assigned Friday 11 March 2011 on orthogonality of sines, due Tuesday 15 March 2011.

    Week of March 14-18, 2011.

  • Did you remember to reset your clocks an hour ahead for Daylight Saving Time?
  • Mid-Term Grades are now available via GoWMU.
  • Monday 3/14: (1) An argument regarding whether or not it matters that the work last week essentially used perfect dipoles, rather than physical dipoles. It turns out we're okay. (2) Development of the Displacement Field vector, D, due to both bound charge densities and free charge densities. Re-writing Gauss' law for D rather than E. Note that unlike E, where curl-E = 0, that curl-D is not necessarily zero, because curl-P isn't necessarily zero -- it isn't for the bar electret, for example. This means that the Displacement Field vector D cannot be written as the gradient of a scalar potential, unlike E.

    Tuesday 3/15: Discussion of situation with Fukushima I Nuclear Plant after tsunami damage in Japan, historical context with Three Mile Island in Pennsylvania and Chernobyl in the former Soviet Union.

    Thursday 3/17: The electric susceptility, chi-sub-e, the permitivity of a material, epsilon-naught × chi-sub-e, and the relative permitivity, (1 - chi-sub-e) = kappa = dielectric constant. Using the Displacement vector D, to find E and P.

    Friday 3/18: For linear dielectrics, since D and P are both proportional to E, then since curl E = 0, then are curl D and curl P also zero? Not necessarily. For example, if one does a closed line integral of P · dl around a path that goes on both sides of an interface between two media, then P will have two values of chi-sub-e, and so the parallel legs won't cancel. By Stoke's theorem, therefore curl P is not zero. In a crystal, it is easier to polarize in some directions than others, so we may get a susceptibility tensor, with 9 terms. For isotropic media (isotropic homogeneous linear dielectric) only the xx, yy and zz diagonal terms survive, and they're all the same.

  • HW6 will be Griffiths Problem 4.17. Due Tuesday 22 March 2011.
  • Quiz 6 will be Griffiths Problem 4.15. And Griffiths Problem 4.18 (illustration on p.185). Due Thursday 24 March 2011.
  • Week of March 21-25, 2011.

    Monday 3/21: Griffiths Example 4.7, pp. 186-188. A while ago we looked at a conducting sphere in a uniform external E-field in the +z-direction. (E = 0 inside, E-field lines must terminate perpendicular to the conducting sphere's surface -- otherwise there is a parallel E-field and the surface charges would still be moving and we wouldn't be in electrostatic equilibrium.) Now we look at a dielectric sphere in the same uniform external E-field in the +z-direction. Have to solve the Boundary Values problem for the potential V. (E inside is parallel to external E-field. E-field lines do not have to end up perpendicular to surface.)

    Tuesday 3/22: Griffiths Example 4.8, pp. 188-190. Put a charge +q at (0,0,d) on the z-axis above the x-y plane. To find the force on the charge if there was a semi-infinite conductor at z < 0, we solved this by method of images by placing a charge -q at (0,0,-d). Now imagine that instead of a semi-infinite conductor, we put a dielectric.

    Thursday 3/24: Energy stored in a capacitor with dielectric. Note that there is more than one way to consider what we mean by the energy to assemble a system -- one may or may not be including the work to "stretch the springs" in the dielectric. Electric force pulling a dielectric slab into a parallel plate capacitor due to real fringe field effects.

    Friday 3/25: Exam 3.

    Week of March 28-April 1, 2011.

    Monday 3/28: Discussion of Electric and Magnetic Forces and Fields of Moving Electric Charges.

    Tuesday 3/29: Lorentz Force. Magnetic Force on a Moving Electric Charge - The Cross Product and Right-Hand Rule (R.H.R.). The Cross Product (or Vector Product) is the exact opposite of the Dot Product (or Scalar Product). Multiplying two vectors together by a cross product gives us another vector (instead of a scalar). And the cross product is not commutative, vector-A × vector-B = - (vector-B × vector-A), so the order is paramount. Using Right Hand Rule to assign directions to x,y,z coordinates. Constant speed, perpendicular constant magnetic force --> Uniform Circular Motion. Cyclotron frequency -- no dependence on the radius (constant angular velocity). If there is a component of the velocity along the B-field direction, get helical paths. Charged particles from the sun directed towards poles -- origins of auroras. Radiation exposure on over-the-poles airline flights. Significant that (a) Mars has only a thin atmosphere and (b) not much magnetic field?

    Thursday 3/31: Discussion of the cyclotron. "Dees" refer to semi-circular (D-shape) magnets. The National Superconducting Cyclotron Lab at MSU. We usually treat the wires in a circuit as having R=0, but they usually are not superconductors. Resistance is a function of temperature. Kammerleigh Onnes 1916 work on extending the R vs. T curve toward T = 0 Kelvin. Discovered Superconductivity, where R=0 identically. High-temperature superconductors.

    April 4/1: Velocity Selector - the Magnetic Force is speed dependent, the Electric Force is not. So we can use an E-field to create an Electric Force to cancel the Magnetic Force on a moving charged particle, such that at the speed v = E / B, the particle travels exactly straight with no net force -- any other speed and the particle is deflected into a barrier. Hence a velocity selector "selects" velocities... Velocity Selector. Mass Spectrometer - different semi-circular paths for ions of different mass but same velocity. Can determine chemicals, molecules, and separate isotopes (same element, different number of neutrons in nucleus, so different mass -- cannot be separated by ordinary chemical means). Mass Spectrometer as Calutron -- detecting or separating isotopes, something that cannot be done by ordinary chemical means. 1895, J.J. Thompson discovers charge and mass of the electron. Griffiths Example 5.2 -- E- and B-fields acting on a charge initially at rest. Get a cycloid trajectory, similar to a point on the rim of a rolling wheel. Next up, electric current. Hall Effect -- a device with no moving electrical parts -- proves that charge carriers in a current carrying wire are negative, not positive.Quiz 8 Take-Home, based on Griffiths Example 5.2 from class, due Tuesday 5 April 2011.

    Week of April 4-8, 2011.

    Monday 4/4: Current carrying wire. I-vector, K-vector, J-vector.

    Tuesday 4/5: Calculating current density J for (a) uniform current, (b) radially dependent current and (c) non-uniform current (!). The Divergance of J-vector is a statement on charge conservation (Continuity Equation). Electrostatics vs. Magnetostatics. The Biot-Savart Law.

    Thursday 4/7: B-field on the z-axis above a circular loop of current. The divergence and curl of B.

    Friday 4/8: Ampere's Law. B-field from a long straight wire -- almost a "circular argument". B-field from an infinite sheet of current -- magnetic analog to the infinite sheet of charge. Open vs. closed coils, how we make coils -- and why we model coils as stacks of circular currents.

    Week of April 11-15, 2011.

    Monday 4/11: Return X3. Ampere's Law and B-field from a long straight Solenoid, Toroidal coil.

    Tuesday 4/12: We're almost to Maxwell's Equations in differential form. Comparisons and differences between E and B -- they seem to be opposites. Still no magnetic monopoles. The Vector Potential A for creating B. Modifying A so as to get cleaner equations. Because A is a vector, ultimately it is not as useful as the scalar potential V is to E.

    Thursday 4/14: Griffiths looks at the 5 equations linking J-vector, A-vector and B-vector -- the 6th equation for symmetry's sake is "not very useful". E-field has discontinuity at a surface charge, likewise B-field has discontinuity at a surface current. A-vector is continuous across the boundary, but the first derivative to the normal component is discontinuous. Multi-pole expansion of the vector potential A. The monopole term must be zero, because (a) we have detected no magnetic monopoles and (b) we therefore designed the vector potential A with no magnetic monopoles in mind. First Day to turn in your Topic 1 Paper.

    Friday 4/15: Exam 4. Second Day to turn in your Topic 1 Paper.

    Week of April 18-22, 2011.

    Monday 4/18: Multipole Expansion of magnetic vector potential A. There is no monopole term. Expect the dipole term to dominate. Magnetic Dipole Moment, m = I a , where a-vector is the area enclosed by the current loop, with the direction taken by the "Mode 2" R.H.R. around the current. Note that if we did find magnetic monopoles, it might be the case that we'd try to make a magnetic dipole moment m = qm d , in a way similar to the electric dipole moment of two charges ±q separated by a displacement vector d . Finding magnetic dipole moment for a current loop which can be made from two perpendicular square loops. Last Day to turn in your Topic 1 Paper. (Unless you had a Draft paper looked at by Dr. Phil.)

    Tuesday 4/19: Griffiths Chapter 6: Magnetization and H-vector. Chapter 7: Ohm's Law. (PDFs of lecture notes.)

    Thursday 4/21: Return X4. Griffiths Chapter 7: Faraday's Law of Induction, Displacement Current correction to Ampere's Law and Maxwell's Equations in various formulations. Superconductors. Once a supercurrent is established, one does not need an internal E-field to keep it going. A superconductor cannot support an internal B-field -- Meissner Effect -- which means that the supercurrents must be on the surface -- i.e., the difference between Exam 4 Problem 2 parts (b) and (c) for s < a. A superconductor can fail, lose its superconductivity if any of the critical parameters are exceeded: critical temperature (Tc), critical current density, critical external B-field.

    Friday 4/22: Chapter 9: Traveling E-M wave solutions to Maxwell's Equations -- for free space and in linear media. And we nearly made it up to Wave Guides, p. 405, which was something of a goal had we not had to adjust the course partway through.