## 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

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 (i.e. Maple).
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).

## Course Contents

### Unit: Electrostatic Potential of Point Charges

#### Potentials

Prerequisite Ideas Introduction

• GEM § ix-xv

"Write down the electrostatic potential due to a point charge"; write down the gravitational potential due to the earth. 10 min

Fields concept (Lecture)

1. scalar fields are a number at every point in space
2. think of temperature as an example
3. electrostatic and gravitational potential are other examples
• GEM § 2.3.2
4. we bring in a voltmeter with attached wires and actually point to various points in space stating that it would measure a value at every point in space. We return to this visual aid often, but at some stage (perhaps this first day) it is important to point out two things:
• How a voltmeter actually works and that you have to set the zero of potential somewhere.
• The fact that voltmeters do NOT work the way you theoretically want them to may be an issue, especially for experimentalists. Nevertheless, we have found that STUDENTS often miss the fact that what a voltmeter measures and what you mean by electrostatic potential have anything to do with each other, even in principle.
• GVC § Voltmeters

#### Superposition

This section can follow “The Distance Between Two Points”

• Lecture
1. electrostatic potentials and gravitational potentials satisfy the superposition principle. It would have been very difficult for us ever to have developed the field of physics if this were not the case.

#### Two Charges (without Power Series)

• If you feel like your students need some physics right away, part 1 of the next activity now. You can then come back and do part 2 of the activity after you have covered power series (or skip it if you are not covering power series). Alternatively, skip this whole activity now and cover it immediately after power series.

#### Power Series

• Intro lecture 15 min
1. introduce especially language for coefficients, order of term, etc.
2. derive derivative formula for coefficients (most students know this formula, but they don't remember the derivation).
• Properties of Power Series (Lecture) 15 min

### Unit: Potential Due to Continuous Distributions

#### dr(vector)

• Curvilinear Coordinates
1. Lecture Intro with “overheads” (lecture)
2. Discussion of math vs. physics conventions (switch $\theta$ and $\phi$) (lecture)
• GEM § 1.4
• $d\Vec{r}$ intro lecture
1. Draw pictures of $\Vec{r}$ and $\Vec{r}$ + $d\Vec{r}$. Find $d\Vec r$ in rectangular coords 5 min
• scalar line integral (lecture)
1. do an example or two. Use $ds = |d\Vec r|$

#### Calculating Potentials

• Series expansion of potential due to a ring (Extension of 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: Electric Field

#### Derivatives of Scalar Fields

• Partial Derivatives (lecture)
• GEM § 1.2.2
• directional derivatives (lecture)

#### Electric Field

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

### Unit: Electrostatic Energy

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