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

1. For students to develop conceptual and geometric understandings of gravitational and electrostatic potentials and fields, including geometric understanding of vector and scalar fields.
2. 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.
3. 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).
4. 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.
5. For students to consider symmetry in making calculations and as part of sense-making activities.
6. 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).
7. 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.

Course Content

Unit: Potentials from Discrete Sources

• QUIZ
• Coulomb's Law
• 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
• Vectors
• Kinesthetic Activity: Star Trek
• Dot Products
• 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
• Electric Field Vectors
• Small Group Activity: The Ring 2 (E)

Unit: Derivatives of Fields

• QUIZ
• Surfaces Activity: The Hillside
• Small Group Activity: Navigating a Hill
• Kinesthetic Activity: Acting out the Gradient
• Directional Derivatives (Optional)
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

• 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

Boundary Conditions

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

Energy for Continuous Distributions

• Energy for Continuous Distributions (lecture)

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