In both lab and lecture we will also consider the application of these principles in the context of basic electronics.
Why do the homework?
I can't emphasize enough the importance of working the problems. In some of your classes homework is primarily evaluative; the point is for you to demonstrate what you've learned from the readings and lectures. In physics the homeworks are primarily instructional; you learn physics primarily by doing working problems. You must work the problems, think about the results, and understand any mistakes you've made if you wish to attain the type of understanding of the subject required of a working physicist. In at nutshell: If you can't work problems you don't know physics. I (or a grader) will grade the problems, and I'll hand out solutions. I encourage you to read the solutions and understand any mistakes immediately. If it doesn't make sense, ask me about it right away---don't wait until right before an exam.
If you've got a compelling reason why you need an extension, come talk to me in advance. I will not grant a homework extension without penalty if you ask for it on the day the homework is due, so don't ask for one. [If you need such a last-minute or post-facto extension due to extenuating circumstances (e.g. death in the family, sudden illness, travel problem), you'll need to have the Dean of Students or your Class Dean formally make such a request to me and suggest a rescheduled due date. You should also take this route if you need an extension but you don't want to tell me why (say, it's for personal or legal reasons). If you explain your reason to a Dean and the Dean tells me it's OK, that's good enough for me.] In general, though, life will be easier for you and for me if you get into the habit of doing your best to finish the problem set on time and handing in as much as you've been able to complete by the deadline.
The formal lab writeups are due one week from the day the lab was performed, unless otherwise noted. Late labs (and rewrites) will be penalized 10% per day late (including weekends). In any case, each lab not performed or lab report not submitted will automatically cost at least a full letter grade in the final course grade.
The midterms will be scheduled during lab time, dates listed in the table below. In the unlikely event you have a conflict with these days, let me know immediately.
The College requires that all written work for a course except for a final be submitted by 5 pm on the last day of classes. The physics department takes this deadline seriously. After that day/time, no homework or lab reports will be accepted, nor will I conduct exit interviews for labs.
The roles of lectures and textbooks
Lecture will not be a regurgitation of the text, a summary of all you need to know for the course, or a how-to guide for the homework. Rather, I'll try go deeper into selected points. In lecture I'll cover material and do demonstrations related to the readings, but I won't feel obliged to be comprehensive in those places where I feel the text is adequate and I may focus only on a few points that I feel are particularly interesting or subtle. You shouldn't expect to understand what's going on without close study of the readings, and you should come to class with questions you have on the readings. Further, after we settle into the semester a bit, I expect the classes will become less lecture-oriented and more participatory; it will be difficult to reap the maximum benefit from that format if you're not sufficiently prepared to fully participate.
For the problems you can't solve, talk to classmates, attend the problem sessions, or ask me. When you ask me, either try to give you just enough of a hint to get you through, or I'll guide you through the problem with a series of leading questions. I'll never just tell you how to do it. If you run out of time and don't finish the set, start earlier next week. When the solutions come out, look over them right away, before you've forgotten all of the points you were confused about. You think you'll just get clear on it before the next exam, but there's never as much time as you think.
On the other hand, if you find the class too slow for your liking, if you have questions that you aren't getting answers to, if you'd like more detail, if you are frustrated that we aren't digging deeply enough, if you crave more applications, come talk to me. I'm very happy to provide you with additional materials or explanations that will will stimulate you and challenge you at whatever level you can handle.
One word of warning: Amherst College students tend to have lots of extracurriculars of all types. I support this (enthusiastically), and I am occasionally willing to be flexible to facilitate your participation in range of activities, but don't let your extracurriculars overshadow your academics. If you become concerned that your courses are getting in the way of your extracurriculars, you definitely have the wrong mindset. Remember why you're here.
If circumstances in your life beyond the class are the problem, you can come talk to me, but also talk to your class Dean.
Key derivations / chains of logic / results to commit to memory
|1. September 2||Electrostatics I
Sept 3: Introduction to the course
General course information. Brief overview of history and significance of electromagnetism.
[Handouts: Course information sheet, first problem set, lab manual ]
Sept 4: Electrostatics: charge, force and energy
Properties of electric charge: quantized and conserved. Coulomb's law. Brief review of conservative forces and the connection between conservative force, work and potential energy (recall: gravity). Potential energy of an electrostatic system.
[Handouts in lab: Student information sheet, diagnostic questionnaire, page on triboelectricity]
Sept 6: Electrostatic potential energy
Calculate the work to bring two point charges together using Coulomb's law. Define potential energy. With superposition, obtain work to assemble a system of charges and an associated potential energy. Write down expression for potential energy of 1D crystal.
[Handout: How van de Graaff generators work]
Sept 7: Stability of matter / Electric fields
Calculate energy of a 1D crystal. While it has lower potential energy than segregated charges, energy can be further decreased by shrinking collapsing + and - charges on each other. Stability of matter problem: why is matter made up of compact charges stable? (Earnshaw's theorem generalizes this beyond our crystal). Resolution is QM.
Define electric field via force. Auxiliary variable now, but will later be a dynamical, local object. Visualizing electric fields: arrows, field lines.
Read: Purcell, 1.1-1.8
|Lab 0: Questionnaires, Lab overview, and Lab 0.||EI|
|2. September 9||Electrostatics II: Electric Fields
Sept 9: Calculating electric fields
Recap on field lines as visualization tool. E fields for continuous charge distributions. EXAMPLE: E field of electric dipole along dipole axis. EXAMPLE: Field of a uniform line of charge.
Sept 10: Calculating electric fields II (more examples)
EXAMPLE: E field of uniform ring of charge on symmetry axis. EXAMPLE: E field of uniform shell of charge -- Shell Theorem.
Sept 11: Electric flux and Gauss's law
Motivation--why Gauss's law when we've got Coulomb's law?: Recap advantage to thinking directly about fields rather than the sources (charges) which set up the fields. Given the fields, what can we deduce about the sources?
Define flux, first through a patch then as an integral over a general surface.
State and prove Gauss's law: (i) single point charge and spherical surface, (ii) single point charge and arbitrary surface.
[Handout: Supplemental notes on the derivation of the shell theorem]
Sept 13: Gauss's law
Finish proof of Gauss's law: (ii) single point charge and arbitrary surface, (iii) arbitrary charge distribution and arbitrary surface. EXAMPLE: Use Gauss to prove shell theorem. EXAMPLE: Field of a line of charge. EXAMPLE: Field of a sheet of charge. EXAMPLE: Field around two sheets of charge.
Read: Purcell, 1.9-1.15
solution to dipole problem
|Lab 1: Simple DC Circuits||EI|
|3. September 16||Electrostatic Energy and Electric Potential (I)
Sept 16: Energy of electric fields (I)
Questions from class about Gauss's law. Force on a layer of charge.
Sept 17: Energy of electric fields (II)
Finish force on a layer of charge. Potential energy of two sheets of charge. Electrostatic energy per unit volume for this configuration (which generalizes to all configurations). Conceptual issue: is the potential energy to be associated with the charges or with the field?
Sept 18: Electric potential
Potential defined as the work per unit charge to move a charge from a reference point to a final point. Write as a line integral of E. E is a central force, so line integrals are path independent. Because of path independence, the scalar potential has a single value at each point in space and is a scalar field. Equivalently, line integral of E around any loop is zero, which implies E is a conservative force.
Visual representation of potential in terms of surfaces of constant potential (equipotential surfaces). Equipotential surfaces are normal to E fields. Analogous to contour maps, where potential is the height, E is vector in direction of steepest descent.
Sept 20: From potential back to E field
Going from potential to E field: differentiating to get a vector. Define gradient of a field, illustrate with simple examples. From Taylor series, show that gradient gives a vector that points in direction of steepest ascent, has magnitude given by derivative of function in that direction. Geometrically, the gradient of a function is orthogonal to constant surfaces of that function.
Relate E and the potential via the gradient. Superposition holds for potentials. Expression for potential for continuous charge distributions.
Read: Purcell 2.1-2.6
|Lab 2: Simple Resistor Networks||Formal|
|4. September 23||Electric Potential (II) / Understanding Electrostatics
Sept 23: Calculating the potential / Local formulations
EXAMPLE: potential of uniform ring of charge, on symmetry axis. EXAMPLE: line of charge. (Illustrates problem of divergence of potentials for infinite charge distributions.) EXAMPLE: electric dipole potential.
Can we formulate our relationships between potential, charge density, and E field locally?
Sept 24: Local formulation of Gauss's law
Derive local form of Gauss's law using a small surface. Define divergence of a vector field. Prove divergence theorem. Obtain explicit expression for divergence in cartesian coordinates using rectangular boxes.
Sept 25: Examples of divergence, and Poisson's equation
EXAMPLE: sphere of uniform charge density.
Use relation between E field and potential with Gauss's law to obtain local relation between potential and charge density: Poisson's equation. Define Laplacian of a field. Introduce Laplace's equation. Three results for functions which obey Laplace's equation (harmonic functions): Mean value theorem, Earnshaw's theorem, no equilibrium for charged particles in empty space due to electrostatic fields. Set up calculation for proof of mean value theorem.
Sept 27: Three results for harmonic functions
Prove mean value theorem: explicit calculation for point charge, then invoke superposition. Earnshaw: proof by contradiction. Prove no stable equilibrium points (although UNSTABLE equilibrium points may abound). Problem for stability of matter!
Can we formulate path independence of line integrals of a vector field locally? Yes...
Read: Remainder of chapter 2.
|Lab 3: Semiconductor Diodes||EI|
|5. September 30||Electrostatics, Conductors and Capacitors
Sept 30: Path independence locally
Define circulation and comment on its geometrical meaning. Note path independence implies zero circulation for all loops. Define curl as the circulation per unit area at a point. Claim: path independence if and only if curl is everywhere zero. Obtain expression for curl in terms of cartesian coordinates by taking the appropriate tiny loops and surfaces.
Oct 1: Wrapping up path independence (and review)
Recap of how to obtain expression for curl in cartesian coordinates. Summary/recap of connections between path independence, curl, and gradients (in the form of a triangle). A vector with zero curl is expressible as a gradient of a scalar, and if a function is the gradient of a scalar it has zero curl. Stokes theorem. Use Stokes theorem to prove that zero curl everywhere implies path independence of line integrals. Review of course to date: lightning recap of main ideas and results.
Oct 2: Electrostatics on, in, and near conductors
[Started with some review questions prior to the exam.] Properties of conductors, insulators and semiconductors. Facts about conductors: (1) E=0 inside conductor. (2) E perpendicular to surface of conductor at the surface. (3) Conductor is an equipotential surface. (4) Gauss's law relates surface charge density to E field at surface. (5) Net charge resides on surface of a conductor.
Oct 4: Boundary value problems
Recap facts about electrostatics on and in conductors. Specifying boundary conditions. Boundary value problems. Claim existence of solutions. Prove uniqueness of solutions using Earnshaw's theorem. E=0 inside of a hollow conductor.
Read: Purcell, Chap. 3
|Lab: Exam #1||na|
|6. October 7||Conductors, capacitors and
boundary value problems / DC Circuits
Oct 7: Solving potential problems
[Some initial questions from the class: Why does free charge accumulate on the surface of a conductor? If there are multiple cavities in a conductor, is the potential the same in each of them?] Recap of the principles of superposition and the uniqueness theorem. Using Gauss's law to solve for the charge in hollow conductors. Using the principle of superposition to obtain the potential for concentric spherical shells of charge. [Question: Does the principle of superposition hold for conductors?] Problem: point charge and conducting plane. Solution via image charge (method of images) and uniqueness theorem.
Oct 8: More method of images / capacitors
Solve problem of a point charge near a grounded sphere using method of images. Comments on solving boundary value problems numerically using relaxation method. Define capacitance. Capacitance of an isolated sphere. Parallel plate capacitor: solve for the E fields, obtain capacitance. Example: Capacitance of concentric spherical shells.
Oct 9: More about capacitors / Current in circuits
Recap on the calculating the charge distributions on surfaces of parallel plate capacitors. Capacitors in series and in parallel. Energy in capacitors. DC circuits: general discussion. Define current, current density.
Oct 11: Currents and Ohm's law
Calculate drift velocity of electrons in a typical household circuit: very small! Local formulation of conservation of charge. Ohm's law in terms of current density. Translating Ohm's law to V=IR. Resistance of a uniform wire.
Read: Purcell, Chapter 4
addendum to 4.30
|Lab 4: RC Transients and Sinusoidal Response||Formal|
|7. October 14||DC Circuits
Oct 14: Fall Break!
Oct 15: Fall Break!
Oct 16: Current and resistance: microscopic and practical
Microscopic model of current: Ohm's law and expression for conductivity in terms of model parameters. Caveat: collision time seems too long in real metals. Adding resistors in series and parallel. Seval comments about conductors and E-fields. Cliffhanger: Kirchoff's laws necessary for analyzing complicated networks.
Oct 18: Kirchoff's laws and voltage sources
Kirchoff's laws and the physics behind them. Voltage sources: pumps of electric current. Charged capacitor + my hand as a simple model of a voltage source. Kirchoff's law for loops with voltage sources.
Read:Purcell 5.1-5.4, Chap. 6
Problems: (no new problems this week)
|Lab 5: Faraday's Law and Inductors in Circuits||EI|
|8. October 21||Circuits / Magnetostatics
Oct 21: Analyzing circuit loops
Loop method. Branch method. RC circuits: loop equations and solutions. Charging the capacitor. Power, both generally and for a resistor.
Oct 22: Magnetostatic fields: history and phenomenology
Capsule history of magnetism. Basic phenomenology of magnetism. Magnetic force on charged particles. Magnetic force on current-carrying wire. Current loops: force and torque in uniform magnetic fields.
Oct 23: Forces and torques due to static magnetic fields
Torque on a current loop in a uniform B field. Magnetic moment defined. Energy of a current loop in a B-field--analogous to electric dipole in an external E-field. Motion of charged particle in uniform B field.
Oct 25: Applications of magnetic fields / Sources of magnetic fields
Velocity selector. Mass spectometer. Thompson's discovery of the electron. DC motor. Biot-Savart law: statement and general features. Biot-Savart law for volume current distributions.
Read: Purcell Chap. 6
|Lab 6: LC Circuits and Resonance||Formal|
|9. October 28||Magnetostatics
Oct 28: Applications of Biot-Savart Law
B field around an infinite wire. B field due to a curved segment of wire. B field along the symmetry axis of a current loop.
Oct 29: Magnetic dipoles and solenoids / Magnetostatics vs. electrostatics
More on magnetic dipoles and magnetic dipole moments. B field of an infinite solenoid on the symmetry axis. Parallels between our progress in magnetostatics vs. our progress in electrostatics.
Oct 30: Deriving Ampere's law
Start derivation of Ampere's law in integral form.
Nov 1: Deriving Ampere's law (cont'd) / Using Ampere's law
Complete derivation of Ampere's law in integral form. Obtain statement of Ampere's law in differential form. Application: B field of infinite wire (inside and outside). B field of a sheet of current. Gauss's law for magnetic fields, differential and integral forms.
Read: Purcell Chap. 6
|Lab 7: RLC Circuits||EI|
|10. November 4||Magnetostatics / Faraday's law
Nov 4: Solenoids / Potentials for Magnetic fields?
Magnetic field of ideal solenoid using Ampere's law. Maxwell's equations for magnetostatics. Can we write potentials for B fields? Pertinent observations: (1) Line integrals of B fields are path dependent, so no scalar potential. (2) Relation between conservative forces and path independence. Magnetic forces do no work!
Nov 5: Vector potential
Lack of path independence precludes a scalar potential, but a vector potential can be formulated. Make div B = 0 automatic by writing B = curl A. Plug into Ampere's law to relate A to J. Gauge invariance permits simplifying assumption: div A = 0. Resulting equation relating A and J is a Poisson's equation for each vector component. Can write integral relation between J and A, just as for rho and phi in electrostatics. Final observations.
Nov 6: Motional EMFs
Magnetostatic triangle. Recap about EMFs. Motional EMF: conducting rod moving through a uniform B field. Closed circuit moving through a uniform B field at constant v. Magnetic flux. Flux rule for motional EMF in this simple case. Start the generalization of the flux rule for arbitrary wire loops in nonuniform fields.
Nov 8: Faraday's Law
Generalize flux rule to arbitrary wire loops in nonuniform fields. Observe that there is no new physics here. Faraday's "three experiments". Universal flux rule. Force on stationary charges leads Faraday to propose: changing magnetic flux induces electric field. Faraday's law, integral and local forms. Observe: this new E field has a curl!
Read: Purcell Chap. 7
|Lab 8: Transistors||EI|
|11. November 11||Faraday's Law / AC circuits
Nov 11: Exam Review and Faraday's Law
Recap of Faraday's law. Contrast Faraday electric field with electrostatic field. Compare of Faraday electric field equations with magnetostatic field equations: can use same techniques to solve problems. Lenz's law.
Demonstrate proportionality between current in one wire and magnetic flux through another. Define mutual inductance. Write (but don't yet prove) Neumann formula.
Nov 12: Inductance and RL circuits
Derive Neumann formula.
EXAMPLE: mutual inductance of coaxial solenoids.
EXAMPLE: self-inductance of solenoid.
RL circuits: Kirchoff's law and solution.
Nov 13: RL and LC circuits
General solution for RL circuit. Charging the inductor. Energy stored in inductors and magnetic fields. Comparison between LC oscillator and mass-on-spring system.
Nov 15: Damped RLC circuits
Analog with mechanical system with viscous drag. Solution to Kirchoff's law equation. Q factor.
Read: Purcell Chap. 7 and 8
|Lab: Exam #2, 2-5 pm||na|
|12. November 18||AC Circuits / Maxwell's equations
Nov 18: Sinusoidally-driven R, L and C
General solution for RLC circuits. Overdamped and critically damped cases. Obtain expressions for the current through R, L and C when driven by a sinusoidal voltage source. Write expressions for I and V in terms of sinusoidal functions and in terms of phasors. Power (instantaneous and average).
[Handout: Magnetic Forces Doing Work? AJP 42 (1974) 205]
Nov 19: Driven RLC circuits and complex numbers
Recap relation between current and voltage for R, L and C. Current-voltage relation in series RLC circuit. Resonance. Power dissipation at and away from resonance. Width of resonance peak in terms of Q.
Complex numbers: Definition of i, complex numbers. Complex conjugate. Addition, subtraction, multiplication and division of complex numbers. Graphical representation of complex numbers.
Nov 20: Complex numbers in AC circuits
Polar representations of complex numbers. Euler relation: statement and proof. Multiplication and division in polar rep. Physical quantities as real parts of complex numbers. Trig identities using complex numbers. Driven inductor using complex numbers. Complex admittance and impedance defined.
Driven capacitor using complex numbers. Impedance of capacitor. Generalized Ohm's law. Driven RLC circuit using complex number approach. Impedances add like resistances in DC circuits.
Nov 22: Complex numbers approach to AC circuits / Completing Maxwell's equations: Displacement current
Driven RLC circuit: Connect complex number expression for current to relation derived using phasors. Another driven RLC circuit example.
Completing Maxwell's equations: recap of E&M equations. Inconsistent! Inconsistency appears when (1) taking divergence of Ampere's law or (2) applying Ampere's law to a charging capacitor circuit. Maxwell's reasonable fix: modify Ampere's law by adding extra term suggested by continuity equation. This is displacement current, and it works: changing E field induces a B field.
Read:Purcell, Chap. 8 and 9
|Lab 8: AM radio receiver||Formal|
Read: Purcell, Chap. 9 and 10
Problems: see previous week
|13. December 2||Maxwell's equations / Electric fields in matter
Dec 2: Displacement currents / Maxwell's equations
Dec 3: Solutions to Maxwell's equations: light as electromagnetism
Dec 4: Electromagnetic waves / Dielectrics and dipoles
Dec 6: Maxwell's equations in dielectrics
Read: Purcell, Chap. 10
|Lab 9: none!||?|
|14. December 9||Electric fields in matter
Dec 10: (Last day)
Dec 11: Reading period
Dec 13: Reading period
|Lab: Exam review or bonus lecture||EI|
|December 17||Final exam: Dec. 17, 9-12 am, Merrill 4
Area Seminars and colloquia