Table of Contents

### Course Overview

The Static Fields Paradigm uses the contexts of electrostatics and magnetostatics to teach a variety of fundamental physics principles.

### Course Goals

These are the combined goals from the Symmetries and Vector Fields 3-week Paradigms

- For students to develop
**conceptual and geometric understandings of gravitational and electrostatic potentials and fields**, including geometric understanding of vector and scalar fields. - For students to
**compute potentials and fields from distributions of sources**, to calculate**fields from potentials**, and to calculate**changes in potential from a field**using vector calculus. - For students to be able to move between
**algebraic and diagrammatic representations**of these fields, including the use of computer visualization tools (*e.g.,*Mathematica). - For students to learn how to calculate potentials and fields due to both
**discrete and continuous distributions**, and to be able to handle non-uniform densities. - For students to consider
**symmetry**in making calculations and as part of sense-making activities. - To develop the
**mathematical tools**needed to make these computations, including vector algebra, dot products, cross products, gradient, line integrals, and**power series expansions**(especially using power series expansions to make approximations). - For students to develop skills for
**communicating**their physics ideas with verbal and mathematical language (group work, class presentations, writing assignments).

- 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

## Course Content

### Unit: Potentials from Discrete Sources

Some nice review tutorials from Harvey Mudd College

**Hour 1: Electric Potential**- QUIZ
- Coulomb's Law

**Hours 2-3: Potential Due to Point Charges**- Superposition
*Small Group Activity: Visualizing Equipotential Surfaces*

- Equipotential surfaces

**Hour 4: Power Series**This hour is likely to move to Energy and Entropy in the future.- Power Series Approximations
*Small Group Activity: Calculating Coefficients for a Power Series*

- Properties of Power Series

**Hour 5: The Distance between Two Points**- Vectors
*Kinesthetic Activity: Star Trek*

- Dot Products

**Hour 6: Two Point Charges**- Dipoles
*Small Group Activity: Electric Potential for 2 Point Charges (with Power Series)*

**Hour 7: Differential Vector Elements**Math Bits??- The Position Vector
- The
*dr*Vector*Small Group Activity: Vector Differentials*

### Unit: Continuous Charge Distributions

**Hour 8: Line Integrals**Math Bits??- QUIZ
- Line Integrals
*Small Group Activity: Boysenberry Patch*

**Hours 9-10: Integrating Charge Densities**Math Bits??- Charge Density
*Kinesthetic Activity: Acting Out Charge Densities*

- Cross Product
*Small Group Activity: Finding dA and dV*

- Total Charge
*Small Group Activity: Total Charge*

### Unit: Potentials Due to Continuous Distributions

**Hours 11-12: Calculating Potentials**- Find the surface area of a cone
- SQBQ write $d\vec r$ in 3 different coordinate systems
- —— Old ——
- Potential Integrals
*Small Group Activity: The Ring 1 (V)*

- Approximating Integrals with Power Series
- Potential Due to Finite and Infinite Lines

**Hours 13-14: Superposition of Electric Fields**- Electric Field Vectors
*Small Group Activity: The Ring 2 (E)*

### Unit: Derivatives of Fields

**Hour 15-16: Gradient**Math Bits??- QUIZ
- The Gradient
*Surfaces Activity: The Hillside**Small Group Activity: Navigating a Hill**Kinesthetic Activity: Acting out the Gradient*

- Directional Derivatives (Optional)
- Electric Field as Gradient

**Hour 17: Flux**- Electric Flux
*Kinesthetic Activity: The Concept of Flux**Small Group Activity: Calculating Flux**Mathematica Activity: Visualizing Electric Flux*

**Hour 18: Divergence**Math Bits??- The Divergence
*Small Group Activity: Visualizing Divergence*

- The Divergence Theorem
- Gauss's Law (Differential Form)

**Hours 19-20: Gauss's Law**- Gauss's Law (Integral Form)
*Small Group Activity: Using Gauss's Law*

**Hours 21-22: Conservative Fields**- What Fields Are Conservative?
*Small Group Activity: Counting Paths*

- Vector line integrals
- QUIZ
- Relating Work to Potential
*Small Group Activity: Work*

- Finding Potentials from Fields
*Small Group Activity: The Murder Mystery Method*

### Unit: Laplace's Equation

**Hours 23-25: Solving Laplace's Equation**- Second Derivatives
- The Laplacian
- Conductors
- Separation of Variables
*Small Group Activity: Boundary Value Problem*(New activity)*Computer Activity: Relaxation Technique*(Optional/Computation course)

### Unit: Electrostatic Energy

**Hour 26: Energy**- Electrostatic Energy
*Kinesthetic Activity: Energy of Discrete Charges*

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

**Hour 27: Curl**Math Bits??- The Curl as Circulation
*Small Group Activity: Visualizing Curl*

- The Curl in Curvilinear Coordinates

**Hour 28: Current**- Electric Current
- Current Density
*Kinesthetic Activity: Acting Out Current Density*

**Hours 29-30: Vector Potentials**- QUIZ
- What Is Vector Potential?
*Small Group Activity: The Ring 3 (A)*

**Hours 31-32: Magnetic Fields**- The Biot-Savart Law
*Small Group Activity: The Ring 4 (B)*

### Unit: Ampère's Law

**Hour 33: Ampère's Law**- Symmetry Arguments
*Small Group Activity: Ampère's Law*

**Hour 34: Stokes' Theorem**- Curl Revisited
- Stokes' Theorem
*Small Group Activity: Where Is the Current?*We need a new activity

**Hour 35: Review**

## Related Topics Not Currently Included

### Unit: Conductors

#### Step & Delta Functions (1 hr)

Note that this is now covered in Waves and Oscillations, and also in Periodic Systems

- Reading: GVC § Step Functions–The Dirac Delta Function and Densities
- Step Functions
- Delta Functions

#### Boundary Conditions

These feel like they go really well early in the Capstone to set up E and B fields in matter.

#### 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)