E&M II Learning Goals

These learning goals for upper-division electrodynamics were created by a group of physics faculty from a number of
research areas, including physics education research. Rather than addressing specific content to be covered in a
course (as with a syllabus), this list of *course-scale* learning goals represents what we think students should be able to *do* at this stage of their development as physicists. The list of *topic-specific* learning goals reflects the knowledge and skills that were emphasiszed in the recent transformed E&M II courses at CU Boulder, organized according to their order of presentation in Griffiths. You can also download PDF versions of these course-scale and topic-specific learning goals.

For comparison, see the learning goals articulated for E&M 1 (advanced electro- and magnetostatics).

**Course-Scale Learning Goals**

**Build on earlier material:**Students should deepen their understanding of introductory electromagnetism, junior-level E&M, and necessary math skills (in particular, vector calculus and differential equations).

**Maxwell’s equations:**Students should see the various topics in the course as part of a coherent theory of electromagnetism; i.e., as a consequence of Maxwell’s equations.

**Math/physics connection:**Students should be able to translate a description of a junior-level E&M problem into the mathematical equation(s) necessary to solve it; explain the physical meaning of the final solution, including how this is reflected in its mathematical formulation; and be able to achieve physical insight through the mathematics of a problem.

**Visualization:**Students should be able to sketch the physical parameters of a problem (e.g., electric or magnetic fields, and charge distributions). They should be able to use a computer program to graph physical parameters, create animations of time-dependent solutions, and compare analytic solutions with computations. Students should recognize when each of the two methods (by hand or computer) is most appropriate.

**Organized knowledge:**Students should be able to articulate the important ideas from each chapter, section, and/or lecture, thus indicating how they have organized their content knowledge. They should be able to filter this knowledge to access the information they’ll need to solve a particular physics problem, and make connections between different concepts.

**Communication:**Students should be able to justify and explain their thinking and/or approach to a problem or analysis of a physical situation, in either written or oral form. Students should be able to understand and summarize a significant portion of an appropriately difficult scientific paper (e.g. an*AJP*article) on a topic from electromagnetism; and have the necessary reference skills to search for and retrieve a journal article.

**Problem-solving techniques:**Students should be able to choose and apply the problem-solving technique that is appropriate for a particular situation (e.g., whether to use the integral or differential forms of Maxwell’s equations). They should be able to apply these methods to novel contexts (i.e., solving problems that do not map directly to examples in a textbook), indicating how they understand the essential features of the technique, rather than just the rote mechanics of its application.

**…7a. Approximations:** Students should be able to effectively use approximation
techniques, and recognize when they are appropriate (e.g., at points far away
or very close to the source). They should be able to
decide how many terms of a series expansion must be retained to find a solution
of a given order, and be able to complete a Taylor Series to at least
two terms.

**…7b. Symmetries:** Students should be able to recognize symmetries, and be able
to take advantage of them when choosing the appropriate method of solution
(e.g., correctly applying the Maxwell-Ampere law to calculate the magnetic
field of an infinitely long wire).

**…7c. Integration:** Students should be able to write down the line, surface or
volume integral required for solving a specific problem, and correctly follow
through with the integration.

**…7d. Superposition:** Students should recognize that – in a linear system
– a general solution can be formed by the superposition of multiple
components, and a specific solution found by applying appropriate boundary
conditions.

**Problem-solving skills:**Students should be able to draw on an organized set of content knowledge (**LG#5**), and apply problem-solving techniques (**LG#7**) with that knowledge in order to carry out lengthy analyses of physical situations. They should be able to connect all the pieces of a problem to reach a final solution. They should recognize the value for learning the material of taking wrong turns, be able to recover from their mistakes, and persist in working towards a solution even though they don’t necessarily see the path to that solution when they first begin the problem. Students should be able to articulate what it is that needs to be solved for in a given problem, and know when they have found it.

**Expecting and checking solutions:**When appropriate for a given problem, students should be able to articulate their expectations for the solution, such as the magnitude or direction of a vector field, the dependence of the solution on coordinate variables, or its behavior at large distances. For all problems, students should be able to justify the reasonableness of a solution (e.g., by checking its symmetry, looking at limiting or special cases, relating to cases with known solutions, dimensional analysis, and/or checking the scale/order of magnitude of the answer).

**Derivations/proofs:**Students should recognize the utility and role of formal derivations and proofs in the learning, understanding, and application of physics. They should be able to identify the necessary elements of a formal derivation or proof; and be able to reproduce important ones, including an articulation of their logical progression. They should have some facility in recognizing the range/limitations of a result based on the assumptions made in its derivation.

**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 problems, be able to articulate where they are experiencing difficulty, and take action to move beyond that difficulty (e.g., by asking thoughtful, specific questions).

**Topic-Specific Learning Goals**

**Griffiths
Ch 7: (Maxwell’s Equations, EMF, inductance, boundary conditions)**

Students should be able to:

1.
…apply
Stokes’ Law and the Divergence theorem, and be able to use them to convert 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?)

2.
…write
down, and explain in words and pictures, the full set of Maxwell's Equations.
This includes recognizing which are relevant for solving a given problem, and
correctly using them to generate a solution. In particular, students should be
able to use the integral form of Faraday’s Law to determine induced electric
fields; and the Maxwell-Ampere law to find induced magnetic fields.

3.
…determine
the total resistance or conductivity of a material for sufficiently symmetric
geometries, calculate the total current flowing, and apply Ohm’s Law (**J** = σ**E**) to relate the current density to the electric
field.

4.
…state the
continuity equation in differential and integral forms, explain how it is an
expression of charge conservation, and understand its implications for quantities such as **J** and **E** in
steady-state situations.

5.
…know
how electromotive force 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).

6.
…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.

7.
…set up
and then solve the differential equations describing time-dependent circuit quantities
(such as current, charge, voltage differences) for simple networks of
resistors, capacitors, and inductors; and with AC or DC drivers.

8.
…work with
complex exponentials in circuit analysis (phasor diagrams), and translate
between complex (imaginary) solutions and physically real quantities.

9.
…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 the fields, and dissipated in resistors).

10.
…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: (Conservation laws for charge, momentum & energy)**

Students should be able to:

** **

1.
…physically
interpret the continuity equation (LG 7.4), and be able to use the divergence
theorem to express it in both integral and differential forms.

2.
…recall
how Poynting's theorem is derived (what assumptions go into it, and the basic
mathematical manipulations), and be able to physically interpret and use **S**, along with the energy density, to
solve problems involving the transfer of energy through electric and magnetic
fields.

3.
…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).

**Griffiths Ch 9 (9.1-9.3): (Electromagnetic
waves, reflection and transmission, dispersion, EM waves in
conductors)**

Students should be able to:

1.
…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 familiar with the usual
elements such as wavelength, wave vector, frequency and angular frequency,
period, phase, polarization, and velocity.

2.
…derive
the traveling wave equation (and thus its solutions) in free space and in
matter, starting from Maxwell's Equations. This involves using the
connections between E and B (and wave speed, wavevector, and polarization
directions) as they arise from Maxwell's equations.

3.
…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.

4.
…find
the boundary conditions for EM waves at the interface of different media,
starting from Maxwell's Equations, and apply them to solve for and interpret
the reflected and transmitted waves, and their intensities (including
understanding and working with the Brewster angle and Snell's Law).

5.
…state
the assumptions that go into simple models for the complex index of refraction,
and the associated equations for the dispersion and absorption of
electromagnetic waves in dielectric and conducting media.

**Griffiths Ch 10: (Potentials, gauge
invariance, retarded time)**

Students should be able to:

1.
…compute
time-dependent fields, given the scalar and vector potentials.

2.
…qualitatively
explain the concept and basic consequences of *gauge invariance*.

3.
…use the definitions of the Coulomb and Lorentz gauges, and be able to state/show how they lead to solvable wave equations.

4.
…state the definition
of *retarded time*, and
compute and interpret potential functions using the retarded time formalism in
cases of very simple geometry.

**Griffiths Ch 11: (Radiation)**

Students should be able to:

1.
…identify
terms in mathematical expressions that correspond to radiation fields.

2.
…explain
and justify both the math and physics of the “three approximations” made when
calculating electric or magnetic dipole radiation.

3.
…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).

4.
…derive
and use expressions for the “radiation resistance”.

5.
…qualitatively
describe and use the basic Larmor formula for radiation from an accelerating
point charge.

**Griffiths Ch 12: (Relativity)**

Students should be able to:

1.
…clearly
describe the two principles of relativity, and their basic consequences (e.g.,
the relativity of simultaneity, Lorentz contraction and time dilation).

2.
…use
Lorentz transformations and four-vector notation to express observables in
different inertial frames.

3.
…know
what makes a quantity *invariant* in
the context of relativity, distinguish it from a *conserved* quantity, and use both concepts to compute relativistic
kinematics problems.

4.
…interpret
and sketch space-time diagrams, and connect them to mathematical formulas.

5.
…use
appropriate transformations to convert electromagnetic fields and potentials between
inertial frames.