Course Goals
Contents: (Click, or scroll down)- What we cover, and why
- Expectations
- Detailed chapter level content goals are at the bottom!
What we cover, and why: Physics 3320 covers advanced topics in electricity and magnetism (E&M). It is the 2nd semester of your second course in E&M (Physics 1120 was the first), but the first course in a true (classical) field theory. Classical electrodynamics (in the form of Maxwell's equations) is one of the most successful physical theories that we presently have. While it is a classical theory (no quantum mechanical Uncertainty Principle here), its conflicts with Newtonian mechanics motivated Einstein's development of Special Relativity. Thus, classical E&M is the first relativistically correct field theory. Maxwell's unification of electricity with magnetism (at first viewed as separate phenomena) and then with light, was the first and grandest example of unification of forces in physics.
For these reasons, along with the sheer mathematical elegance and completeness of the theory, and its extraordinary (uncanny!) agreement with experiment, electromagnetism is an inspiration for the creation of other physical theories including quantum mechanics and quantum field theory, and indeed much of contemporary physics. Further, classical E&M is at the root of a huge number of practical applications. Most of the phenomena of everyday experience, sights, smells, texture, etc. arise from a balance of electromagnetic interactions and quantum mechanics. E&M is essential in understanding the physics behind electric power generation, electronics, optics, communications, (and on, and on!) We view the universe around us primarily via the electromagnetic radiation. Clearly, to understand the physical world, we need to understand electricity and magnetism.
COURSE SCALE LEARNING GOALS
- Math/physics connection: Students should be able to translate a physical description of a junior-level electromagnetism problem to a mathematical equation necessary to solve it. Students should be able to explain the physical meaning of the formal and/or mathematical formulation of and/or solution to a junior-level electromagnetism problem. Students should be able to achieve physical insight through the mathematics of a problem.
- Visualize the problem: Students should be able to sketch the physical parameters of a problem (e.g., E or B field, distribution of charges, polarization), as appropriate for a particular problem. Students should be able to choose and use appropriate computer programs (e.g. Mathematica) to create visualizations of charge/current distributions and potentials; produce animations of time-dependent solutions; compare analytic solutions with computations, and recognize when each of the two methods is most appropriate.
- Organized knowledge: Students should be able to articulate the big ideas from each chapter, section, and/or lecture, thus indicating that they have organized their content knowledge. They should be able to filter this knowledge to access the information that they need to apply to a particular physical problem.
- Communication. Students should be able to explain their thinking and/or approach to a problem or physical situation, in either written or oral form. Students should be able to read and understand a significant portion of a scientific paper (written at the level of, say, an American Journal of Physics article) on a topic from electricity and magnetism. Students should also demonstrate necessary reference skills to electronically search for, locate and retrieve a journal article.
- Problem-solving techniques: Students should be able to choose and apply the problem-solving technique that is appropriate to a particular problem. This indicates that they have learned the essential features of different problem-solving techniques (eg., separation of variables, method of images, direct integration). They should be able to apply these problem-solving approaches to novel contexts (i.e., to solve problems which do not map directly to those in the book), indicating that they understand the essential features of the technique rather than just the mechanics of its application. They should be able to justify their approach for solving a particular problem.
…a. Approximations: Students should be able to recognize when approximations are useful, and use them effectively (eg., when the observer is very far away from or very close to the source). Students should be able to indicate how many terms of a series solution must be retained to obtain a solution of a given order.
…b. Symmetries: Students should be able to recognize symmetries and be able to take advantage of them in order to choose the appropriate method for solving a problem (eg., when to use Gauss’ Law, when to use separation of variables in a particular coordinate system).
…c. Integration: Given a physical situation, students should be able to write down the required partial differential equation, or line, surface or volume integral, and correctly calculate the answer.
…d. Superposition: Students should recognize that – in a linear system – the solutions may be formed by superposition of components.
...e. Derivations/Proofs: Students should recognize the utility and role of formal derivations or proofs in learning, understanding and applying physics. They should be able to identify the necessary elements that make up a derivation or proof. Students should be able to do important derivations, which includes being able to articulate the thread of a derivation/proof throughout. Students should demonstrate some facility in determining the range or limits of applicability for a derived result based on the details of a derivation.
...f. Metacognition: Students should be able to justify their choices in problem solving methods (see LG #4 above) verbally or in writing, and explicitly engage in discussion about their thinking and what helped them learn. (See also LG #6 below) - Problem-solving strategy: Students should be able to draw upon an organized set of content knowledge (LG#3), and apply problem-solving techniques (LG#4) to that knowledge in order to organize and carry out long analyses of physical problems. They should be able to connect the pieces of a problem to reach the final solution. They should recognize that wrong turns are valuable in learning the material, be able to recover from their mistakes, and persist in working to the solution even though they don’t necessarily see the path to the solution when they begin the problem. Students should be able to articulate what it is that needs to be solved in a particular problem and know when they have solved it.
- Expecting and checking solution: When appropriate for a given problem, students should be able to articulate their expectations for the solution to a problem, such as direction of the field, dependence on coordinate variables, and behavior at large distances. For all problems, students should be able to justify the reasonableness of a solution they have reached, by methods such as checking the symmetry of the solution, looking at limits, relating to cases with known solutions, checking units, dimensional analysis, and/or checking the scale/order of magnitude of the answer.
- Intellectual maturity: Students should accept responsibility for their own learning. They should be aware of what they do and don’t understand about physical phenomena and classes of problem. This is evidenced by asking sophisticated, specific questions; being able to articulate where in a problem they experienced difficulty; and take action to move beyond that difficulty.
- Maxwell’s Equations. Students should see the various laws in the course as part of the coherent theory of electromagnetism; ie., Maxwell’s equations and conservation laws.
- Build on Earlier Material. Students should deepen their understanding of Phys 1120 and 3310 material. I.e., the course should build on earlier material, including E&M content and appropriate math (including vector calculus and relevent solution methods for differential equations)
Important comment on preparation:
Physics 3320 is a challenging, upper-division physics course. Unlike earlier courses, you are fully responsible for your own learning. Physics 3320 covers much material you have not seen before, at a higher level of conceptual and mathematical sophistication than you may have encountered in a physics class so far.
Therefore you should expect:
- a large amount of material covered quickly.
- no recitations, and few examples covered in lecture. Most homework problems are not similar to examples from class.
- long, hard homework problems that usually cannot be completed by one individual alone.
- challenging exams.
YOU control the pace of the course by asking questions in class. We tend to speak quickly, and questions are important to slow down the lecture. This means that if you don’t understand something, it is your responsibility to ask questions. Attending class and the homework help sessions gives you an opportunity to ask questions. We are here to help you as much as possible, but we need your questions to know what you don’t understand.
Physics 3320 covers some of the most fundamental physics and mathematical methods in the field. Your reward for the hard work and effort will be learning important and elegant material that you will use over and over as a physics major (and beyond!) Here is what we have experienced:
- most students reported spending a minimum of 10 hours per week on the homework (!!)
- students who didn’t attend the homework help sessions often did poorly in the class.
- students reported learning a tremendous amount in this class.
Specific content goals, organized by chapters:
Pre Ch 7: We expect you to have a working knowledge of the essential ideas and tools of prerequisite courses, including but not limited to Newton's law (!), the Lorentz force law, "canonical" electro-and magneto-statics (like finding and interpreting voltage, E, and B fields - using boundary conditions with Maxwell's equations or Laplace's equation, in simple situations)
In particular, we spent some time during our "review" week talking about the "Griffiths triangles" (Section 2.3.5 and
5.4.2), and the boundary conditions on E and B fields.
Students should be able to...
- …derive (from Maxwell's equations!) our boundary conditions on E and B. You should be able to generate (or interpret) the little "pictorial proof" that E(parallel) is continuous at boundaries (using a line integral and Faraday's law), but E(perpendicular) can/will "jump" if there is a surface charge density (using a little volume integral and Gauss' law), and similarly B(perpendicular) is continuous at boundaries (using a little volume integral and Gauss' law for magnetism), but B(parallel) can/will "jump" if there is a surface current flowing. (using a line integral and Ampere's law). (See Griffiths 2.3.5 and 5.4.2 to review those derivations!)
- ... Calculate and sketch the magnetic field given simple (symmetrical) current distributions from Ampere's law, and/or calculate and sketch the electric field given simple (symmetrical) charge distributions from Gauss' lasw
- ... Set up the appropriate integrals to find E fields from Coulomb's law, or B fields from the Biot-Savart law.
Griffiths Ch 7:
Students should be able to...
- …correctly apply Stokes’ Law and the Divergence theorem, and be able to use them to convert a variety of equations from differential to integral form (and vice-versa), as well as interpret the physical meaning of the resulting terms (e.g., what's a flux? what's a divergence? what does the curl tell you?)
- …write down, and explain in words and pictures, the full set of Maxwell's Equations in vacuum. This includes recognizing which are relevant for solving a given problem, and using them to solve problems with sufficient symmetry. In particular, students should be able to use the integral form of Faraday’s Law to determine induced E-fields, and the Maxwell-Ampere law to determine induced B-fields.
- …determine the total resistance or conductivity of a material for a given geometry (assuming the geometry has sufficient symmetry to make the problem tractable), and be able to apply Ohm’s Law (in the form J = sigma E) to relate the current density to the electric field, and calculate the total current for a given situation.
- …state the continuity equation in both differential and integral form, explain how it is an expression of charge conservation, and understand the implications of this equation for quantities such as J and E in steady state situations.
- …know how EMF is defined, and be able to compute EMF (either motional or Faraday-induced) for a variety of situations (typically with known or easily computed fields).
- …relate mutual and self-inductance to magnetic flux and the resulting induced EMF and currents, and find the mutual and self-inductance for a situation with sufficient symmetry.
Exam 1 covered up to here. Exam 2 can cover any of the above, but focus will be on subsequent material:
- …set up and then solve the differential equations describing time-dependent circuit quantities (such as current, charge, voltage differences) in circuits with either AC or DC drivers (or no driver at all), and simple networks of resistors, capacitors, and inductors.
- …work with complex exponentials as solutions to differential equations in circuit analysis, and be able to translate between complex (imaginary) solutions and physically real quantities.
- …calculate the total energy contained in electromagnetic fields (from the energy densities), as well as compute the power and energy flow in AC circuits (by considering the energy stored in inductors and capacitors, and dissipated in resistors)
- …understand how to translate between E- and B-fields and the auxiliary fields D & H, in terms of the polarization and magnetization of a material, and be able to set up appropriate boundary conditions to determine E, B, D & H at the interface of two different media.
Griffiths Ch 8:
- ... interpret the continuity equation, and manipulate it using the divergence theorem to express it in integral and differential form.
- …recall how Poynting's theorem is derived (what assumptions go into it, and the basic mathematical manipulations), and to physically interpret and use S, along with the energy density, to solve problems involving the transfer of energy through electric and magnetic fields.
- ... correctly determine the direction of S in physical situations, and interpret the sign of the flux integral of S in terms of energy flow.
- …qualitatively use the expression for momentum volume density, and explain its connection to conservation of momentum in EM systems (and similarly for angular momentum volume density)..
(Note that we did NOT cover the stress-energy tensor in detail, so I will not ask you to do problems involving the elements of that "T" tensor.)
- …construct general solutions to the wave equation, in 1-D to 3-D (including superposition solutions which might not look like the canonical "traveling wave"). For traveling waves, students should be comfortable working with the usual elements such as wavelength, wavenumber, frequency and angular frequency, period, phase, polarization, and velocity.
- …derive the traveling wave equation (and thus its solutions) in free space and in matter, starting from Maxwell's Equations. This would include deriving, understanding, and using the connections between E and B (and wave speed, wave vector, and polarization directions) as they arise from Maxwell's equations.
- …interpret and work with plane wave solutions in complex notation, and move back and forth between this notation and the real (physical) wave formulas, as well as apply concepts from previous chapters (S, energy, momentum and angular momentum densities) in the context of EM plane waves.
- …find the boundary conditions for EM waves in free space and in matter, starting from Maxwell's Equations, and apply the correct boundary conditions to solve for and interpret reflected and transmitted waves, and reflected and transmitted intensities (including understanding and working with the Brewster angle and Snell's Law).
NOTE: Midterm #2 Fa '14 is going to stop here. What follows could be on the final, though! - ... extend the previous item to include the case of EM waves normally incident on a conductor. This includes interpreting the complex wave-vector that arises, knowing the relevant appoximations and assumptions we made to derive simple results, and being able to compute, interpret, and use the resulting formulas for reflection and transmission coefficients (even in this case where they are complex!)
Note that the test will not cover 9.4.3 "dispersion" in materials, and we do not cover 9.5 "Wave guides" at all.
- ... compute time dependent fields, given scalar and vector potentials.
- ... qualitatively explain the concept and basic consequences of "gauge invariance" (including basic manipulations of Griffiths Eq 10.7)
- ... know/use the definitions of "Coulomb" and "Lorentz" gauges, and be able to state/show how they lead to solvable wave equations
(Originally, I said exam 2 Fa'13 covers up to here only).
- ... qualitatively interpret "retarded time", and compute and interpret potentials using the retarded time formalism in cases of very simple geometry.
- ….identify terms in expressions which correspond to "radiation fields"
- …explain and justify both the math and physics of the "three approximations" involved in calculating electric or magnetic dipole radiation.
- ... use and explain the mathematical forms of E and B for electric or magnetic dipole radiation fields, including computation or representation of energy flow, intensity, and power, (with angular and radial dependence)
- ... derive and use "radiation resistance" of simple dipole radiators
- ... qualitatively describe, and use in straightforward (nonrelativistic) cases, the basic Larmor formula for radiation from an accelerating point charge.
- …. be able to clearly describe the two principles of relativity, and their basic consequences (including relativity of simultaneity, Lorentz contraction and time dilation)
- ... use Lorentz transformations and four-vector notation to convert relevant observables between inertial frames.
- ... Know what "invariant quantity" means in the context of relativity, distinguish it from "conserved quantities", and use both to compute e.g. relativistic kinematics problems.
- ... be able to interpret and sketch space-time diagrams, and connect them to mathematical formulas.
- ... Be able to use E and B transformations to convert fields between frames. (I won't put this one on the final this year, since our homework on this won't get back to you in time. But it's how we'll wrap up the term!)