## Unit: Gauss's Law

### Flux (1+ hr)

• 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)

• 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.
• Gauss' Law (SGA) - students solve for the electric field due to a charged sphere or an infinite cylinder. Emphasis is made on students making symmetry arguments (proof by contradiction) for using Gauss' Law.

### Divergence

• Definition of divergence (lecture) We follow “div, grad, curl and all that”, by Schey

### Divergence Theorem

• 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

• Differential Form of Gauss's Law: Maxwell's Eq 1 & 3: $\Vec{\nabla} \cdot \Vec{E} = {\rho \over \epsilon_0}$, $\Vec{\nabla } \cdot \Vec{B} = 0$ (lecture)
• (optional) Divergence of a Coulomb field (requires delta functions) (lecture)
• (optional) Electric field lines (lecture)

## Unit: Current, Magnetic Vector Potential, and Magnetic Field

### Vector Potentials (Optional)

• Vector Potential A (lecture) 10 min max This can be just an analogy with electrostatic potential.
• Curl (at least the component definition in rectangular coordinates)
• Derivation of Biot-Savart from Vector Potential (lecture)

### Magnetic Fields

• Biot-Savart (lecture) 15 min (Physics 37: The Biot-Savart Law)
• Comparing B and A for spinning ring (class discussion/lecture - optional) (Physics 40: Comparing B & A for the Spinning Ring

## Unit: Ampère's Law

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

### Curl

• Circulation (lecture)
• Definition of Curl (lecture). We follow “div, grad, curl and all that”, by Schey

### Stokes' Theorem

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

### Differential Form of Ampère's Law

• 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

### Conductors (1/2 hr)

• Conductors (lecture)

## Unit: Conservative Fields

### Conservative Fields

• 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

• Second Derivatives & the Laplacian (lecture)
• (optional) Laplace's Equation (SGA)

## Unit: Energy

### Product Rules

• Integration by Parts (lecture)
• Product Rules (lecture)

### Energy for Continuous Distributions

• Energy for Continuous Distributions (lecture)

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