Table of Contents
Static Vector Fields
The Static Vector Field Paradigm continues the discussion of E&M from the Symmetries & Idealizations Paradigm, focusing on electric fields, magnetic fields, and the magnetic vector potential. This course uses a variety of pedagogical techniques (small group activities, computer visualization, kinesthetic activities, and lecture/discussion) to help students build a multifaceted understanding of these ideas. This course emphasizes extending the integral versions of Maxwell's equations (learned in introductory physics) to the local, differential versions; visualizing vector-valued functions in three dimensions using the computer algebra software Maple; and extending the techniques of vector calculus from rectangular to cylindrical and spherical coordinates. (Catalog Description)
Course Goals
- For students to build conceptual and geometric understanding of current density, magnetic field, and magnetic vector potential and a formal understanding of the relationships between them (using vector calculus)
- For students to understand divergence and curl - formally and geometrically - and the Divergence Theorem and Stoke's Theorem formally and geometrically
- To derive the differential form of Maxwell's equations from the integral form and for students to have link their conceptual understanding with the formalism of Maxwell's equations
- For students to understand Gauss' Law and Ampere's Law and how to make explicit symmetry arguments.
- For students to understand the continuity of electric and magnetic fields across charge/current boundaries.
- For students to understand how energy is stored in electric and magnetic fields, and be able to calculate the energy from sources, fields and potentials.
- For students to come to understand that sources, fields, and potentials are different constructs that address the same phenomena, but are useful in different ways.
Sample Syllabus
Textbook: The Geometry of Vector Calculus—-an introduction to vector calculus, with applications to electromagnetism. One of the Tables of Contents for this online interactive textbook has been specifically designed for this course.
Course 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)
Activities Included
- All activities for Static Vector Fields
