## 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)

### 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