## Symmetries & Idealizations

The Symmetries & Idealizations Paradigm is the first of the paradigms courses. It focuses on topics in electrostatics and magnetostatics with relevant math methods introduced in a "just-in-time" manner. Students are asked to solve complex problems involving electrostatic potential and electric fields using power-series expansions to approximate the behavior of these fields in various regions of space. Through a variety of pedagogical techniques, students learn about fields, potentials, charge densities, curvilinear coordinates, partial derivatives, dealing with vectors in integrands, delta functions, gradients and directional derivatives. A central feature of this course is a sequence of activities that bridges this Paradigm with the Static Vector Fields Paradigm: a set of scaffolded activities designed to help students learn to break complicated problems into more manageable pieces. (Catalog Description)

The course is divided into three units, each of which ends with a group activity that requires the students to use many different types of cognitive resources to solve. Subunits leading up to these summative activities each focus on a specific cognitive resource.

In the first unit, the summative group activity asks students to find the electrostatic potential due to a pair of charges and then to expand that potential in a series valid on an axis of symmetry. In a whole class discussion, students compare and contrast the examples done by different groups, focussing on the role of symmetry in the problem. Subunits address: a review of power series methods and theorems, the geometric interpretation of $|\Vec r - \Vec r'|$, and the geometric implications of the superposition principle.

In the second unit, the summative group activity asks students to find the electrostatic potential due to a ring of charge and then to expand that potential in a series valid on the axis or the plane of symmetry. In a whole class discussion, students compare and contrast the examples done by different groups, focussing on the physical meaning of the terms in the series expansions. Subunits address: integration in curvilinear coordinates, different types of densities. An optional subunit addresses the example of the infinite line charge, exploring the role of series expansions in taking the limit from the finite to the infinite line and also the role of the zero of potential in cases where the charge distribution extends to infinity.

In the final unit, the summative group activity asks students to find the electric field due to a ring of charge and then to expand that field in a series valid on the axis or the plane of symmetry. In a whole class discussion, students compare the results for the electric field to the results for the electrostatic potential. Subunits address: the geometric interpretation of partial derivatives, directional derivatives, and gradients; the relationship between electrostatic potentials, electric fields, electrostatic energy, and electrostatic forces; and the electrostatic energy due to a discrete distribution of charges.

### Course Goals

- 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 (i.e. Maple). - 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).

### Sample Syllabus

## Course Contents

### Unit: Potentials from Discrete Sources

#### Potentials

- Reading: GVC § Idealizations and Symmetries
- Reading: GEM § ix-xv
- Introduction
- Electric Potential (SWBQ: 10 min)

- Fields concept (Lecture: ?? min)

#### Superposition

* This section can follow “The Distance Between Two Points”*

- Superposition (Lecture: ?? min)
- Visualizing Electrostatic Potentials (Maple/Mathematica: 20 min)

#### The Distance Between Two Points

- Reading: GEM § 1.1.1-1.1.2, 1.1.4
- Dot product review (SWBQ: 15 min)

- Magnitudes of vectors (Lecture: 5 min)
- The Distance Between Two Points: Star Trek (Kinesthetic Activity: 20-30 min)

#### Two Charges (without Power Series)

- Reading: GVC § Dipoles–More Dipoles

### Unit: Power Series Approximations

#### Power Series Basics

- Reading: GVC § Power Series–Properties of Power Series
- Power Series (Lecture: 15 min)
- Approximating Functions with a Power Series (Maple/Mathematica)
*30 min* - Properties of Power Series (Lecture: 15 min)

#### Two Charges with Power Series

- Reading: GVC § Dipoles–More Dipoles

### Unit: Continuous Charge Distributions

#### dr(vector)

- Reading: GEM § 1.4
- Curvilinear Coordinates (lecture)
- Scalar Line Integral (lecture)

#### Integrating Charge Densities

- Reading: GEM § 1.1.2, 1.1.4, 1.3.1, 2.1.4
- Acting Out Charge Densities (kinesthetic)
- Defining the Cross Product (SWBQ)
- Cross products and scalar triple products (lecture)

### Unit: Potentials Due to Continuous Distributions

#### Calculating Potentials

- Series expansion of potential due to a ring of charge (Extension of previous SGA)
- Potential due to a finite line (lecture)
- Potential due to infinite line (lecture) (This is a longish lecture and a bit more sophisticated than much of the other material. It is an excellent opportunity to do lots of series expansions and review logarithm rules. Alternatively, it can be left out to save time.)

### Unit: The Electric Field as a Gradient

#### Derivatives of Scalar Fields

- Reading: GEM § 1.2.2
- Partial Derivatives (lecture)
- Curvilinear Basis Vectors (kinesthetic)
- Introducing $d\Vec{r}$ (lecture)
- Gradient (lecture)
- Visualizing Gradient (Maple/Mathematica)
- directional derivatives (lecture) (Optional)

#### Electric Field

- Electric Field - as gradient of the potential (lecture)

### Unit: Superposition of Electric Fields

### Unit: Electrostatic Energy

*This unit could also be effectively placed in several locations in Paradigm: Vector Fields.*

- Electrostatic Energy (kinesthetic)
- Vector line integrals (lecture)

### Activities Included

- All activities for Symmetries & Idealizations