Table of Contents
Unit: Gauss's Law
Flux (20-50 minutes)
- Reading: GVC § Flux–More Flux through a Cube
- Recall Flux (SWBQ) 5 min
- The Concept of Flux (Kinesthetic Activity) 5 min
- Calculating Flux (Small Group Activity–Optional) 30 min
- Visualizing Electric Flux (Maple) 10 min - 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.
- Flux (Lecture, if necessary) Fill in any holes not covered by the activities and class discussions.
Gauss's Law (120 minutes)
- Reading: GVC § Gauss's Law and Symmetry–More Gauss's Law: Cylinders and Spheres
- Gauss' Law -- the integral version (Lecture) 30 min
- Gauss' Law (SGA ) 90 min - 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 (40 min)
- Definition of divergence (Lecture) 20 min
- Visualizing Divergence (Maple Visualization) 20 min Students practice estimating divergence from graphs of various vector fields.
Divergence Theorem (20 min)
- Reading: GVC § 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 (10 min)
- Reading: GVC § Differential Form of Gauss's Law–Electric Field Lines
- 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
Current
- Reading: GVC § Currents
- Acting Out Current Density (kinesthetic)
- Current Density (lecture) 10 min
Vector Potentials (Optional)
- Reading: GVC § Magnetic Vector Potential–Curl
- 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)
Magnetic Fields
- Derivation of the Biot-Savart Law from Magnetic Vector Potential (lecture) 15 min
- (optional) Comparing B and A for spinning ring (class discussion/lecture)
Unit: Ampère's Law
Ampère's Law
- Ampère's Law and Symmetry Argument (Lecture) 20 min
Curl
- Circulation (lecture)
- Visualizing Curl (Maple)
- Definition of Curl (lecture). We follow “div, grad, curl and all that”, by Schey
Stokes' Theorem
- Reading: GVC § 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
Step & Delta Functions (1 hr)
- Reading: GVC § Step Functions–The Dirac Delta Function and Densities
- Step Functions
- Delta Functions
Conductors (1 hr)
- Conductors (lecture)
Boundary Conditions
Unit: Conservative Fields
Conservative Fields
- Reading: GVC § Independence of Path–Finding potential Functions
- Conservative Fields (lecture) (Math 3.5: Independence of Path, Math 3.6: Conservative Vector Fields, Math 3.7: Finding Potential Functions)
- Equivalent Statements (lecture)
Second Derivatives
- Reading: GVC § Second Derivatives,The Laplacian
- Second Derivatives & the Laplacian (lecture)
- (optional) Relaxation Technique for Solving Laplace's Equation (SGA)
Unit: Energy
Product Rules
- Reading: GVC § Product Rules–Integration by Parts
- Product Rules (lecture)
- Integration by Parts (lecture)
Energy for Continuous Distributions
- Energy for Continuous Distributions (lecture)
