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

Fall 2008

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)

Gauss's Law (120 minutes)

Divergence (40 min)

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)

Unit: Current, Magnetic Vector Potential, and Magnetic Field


Vector Potentials (Optional)

Magnetic Fields

Unit: Ampère's Law

Ampère's Law


Stokes' Theorem

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)

Conductors (1 hr)

Boundary Conditions

Unit: Conservative Fields

Conservative Fields

Second Derivatives

Unit: Energy

Product Rules

Energy for Continuous Distributions

Activities Included

Personal Tools