Unit: Gauss's Law

Flux (1+ hr)

Prerequisite Ideas

  • Definition of Flux (lecture & kinesthetic activity)
    1. The Concept of Flux (kinesthetic activity)
    2. If you haven't yet done so, this is the time to introduce the vector surface element $d\Vec a= \hat n da$, pointing out that the only unique direction related to a surface (in three dimensions) is the normal to the surface.
    3. It is helpful to bring in a bunch of rulers or meter sticks for students to use to make a vector field and something like a hula hoop to represent a surface.
    4. since this is a course in electroSTATICS, it is important to avoid examples with time dependence and language like the amount of stuff that “gets through” the surface (“points through” is more helpful language).
    5. emphasize that the dot product only picks up the component of the vector field perpendicular to the surface.
  • Calculating Flux (Small Group Activity)
  • Visualizing Electric Flux (Maple) - plots electric field vectors from a charge in a box and calculates the flux through the surfaces of the box. Leads to a statement of Gauss' law.

Gauss's Law (2 hrs)

Prerequisite Ideas

  • Gauss' Law – the integral version (lecture) 10 min
  1. most students remember something about this law. This makes a nice “recall” SWBQ. Ask the students to write down Gauss's law on their small white boards and then construct the answer on the front board from the bits and pieces that they remember, emphasizing why each of the pieces is important and talking about the geometry of each piece.
  2. emphasize that Gauss's law holds for ANY electrostatic field and ANY (imaginary) surface
  3. most students will look for a real surface in the problem, rather than an imaginary one. Emphasize this. Maybe it would be worth giving them a charge distribution and asking them to draw the Gaussian surface that works. Collect a bunch and then emphasize that they all work.
  4. do the example of Gauss's law for a thick plane of charge, modelling the symmetry arguments involved.

Divergence

Prerequisite Ideas

Divergence Theorem

Prerequisite Ideas

  • Derivation of the Divergence Theorem (lecture). We follow “div, grad, curl and all that”, by Schey. The Divergence theorem is almost a lemma based on the definition of divergence. Draw a diagram of an arbitrary volume divided into lots of little cubes. Calculate the sum of all the fluxes out of all the little cubes (isn't this a strange sum to consider!!) and argue that the flux out of one cube is the flux into the adjacent cube unless the cube is on the boundary.

Differential Form of Gauss's Law

Prerequisite Ideas

Unit: Current, Magnetic Vector Potential, and Magnetic Field

Current

Prerequisite Ideas

Vector Potentials (Optional)

Prerequisite Ideas

Magnetic Fields

Prerequisite Ideas

Unit: Ampère's Law

Prerequisite Ideas

  • Ampère's Law and Symmetry Argument (Lecture) 20 min (Physics 42: The Magnetic Field of a Uniform Planar Current)

Curl

Prerequisite Ideas

Stokes' Theorem

Prerequisite Ideas

  • Derivation of Stokes' Theorem (lecture). We follow “div, grad, curl and all that”, by Schey

Differential Form of Ampère's Law

Prerequisite Ideas

  • Stokes' Theorem (lecture) (Math 3.12: Stokes' Theorem)
  • Differential Form of Ampère's Law: Maxwell Eq. 2 & 4 $\Vec{\nabla } \times \Vec{E} = 0$, $\Vec{\nabla } \times \Vec{B} = \mu_0 \Vec{J}$(lecture) (Physics 41: Differential Form of Ampère's Law)

Unit: Conductors

Step & Delta Functions (1 hr)

Conductors (1/2 hr)

Unit: Boundary Conditions

Unit: Conservative Fields

Conservative Fields

Prerequisite Ideas

  • Conservative Fields (lecture) (Math 3.5: Independence of Path, Math 3.6: Conservative Vector Fields, Math 3.7: Finding Potential Functions)
  • Murder Mystery Method (SGA)
  • Equivalent Statements (lecture)

Second Derivatives

Prerequisite Ideas

Unit: Energy

Product Rules

Prerequisite Ideas

Energy for Continuous Distributions

Prerequisite Ideas

  • Energy for Continuous Distributions (lecture)

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