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## Acting Out Charge Densities : Instructor's Guide

### Main Ideas

• Conceptual understanding of linear, surface and charge density
• The distinction between uniform and non-uniform densities

Estimated Time: 10 minutes

To pretend like they are charges and form various types of charge densities throughout the room.

### Prerequisite Knowledge

Some understanding of mass density is helpful.

None

### Activity: Introduction

Note: It helps if the instructor stands on a chair or table so they are high enough to see all the students. We usually hold a voltmeter and pretend that it is set to measure electric fields.

Prompts: Each of you is a charge:

• Make a linear charge density.
• Make a surface charge density.
• Make a volume charge density.

### Activity: Student Conversations

• Students usually line up in a straight line. An excellent follow-up question is “Does a linear charge density need to be arranged in a straight line?”
• A few students may interpret the word linear to mean that the number of charges in each interval of space is increasing. This is an excellent opportunity to discuss the two different uses of term “linear” - as “one-dimensional” and as “$y=mx$”.
2. A next question might be “Is this a uniform or non-uniform density?”.
• You can then discuss what “uniform” means.
• You can ask the students to make their distribution first uniform and then non-uniform.
3. “Make a surface charge density.”
• Students will spread themselves around the room, or line up two-by-two. You can ask the students whether their distribution needs to be flat.
4. At some point, the idea of idealization should come up. What do we mean by a continuous distribution of charges described by a charge density?

### Activity: Wrap-up

This activity can wrap up with a presentation of the variables used to describe the various types of charge densities ($\lambda$, $\sigma$, and $\rho$). Its also useful to mention the dimensions of these charge densities and conceptually how one would measure them (take a meter stick and count charges).

### Extensions

If this activity works well in your classroom, you might also want to try the activity

at an appropriate time in your course.

This activity is included within a sequence of activities addressing Gauss’s law in integral form. The following activities are part of this sequence and can be used as extensions to this activity.

• Follow-up activities:
• Total Charge: In this small group activity, students calculate the total charge in spherical or cylindrical dielectric shells from charge densities which vary in space.
• The Geometry of Flux Sequence: This sequence of activities addresses the geometry of flux as well as allowing students ample practice in the mathematics which is used to calculate flux through various surfaces.
• Gauss's Law--the integral version: This lecture introduces Gauss's law in integral form and serves as an introduction to the Gauss's Law activity.
• Gauss's Law: This small group activity is the final activity in the Geometry of Flux sequence where students calculate the electric field due to various charge densities in spherical and cylindrical shells.

This is the initial activity within a sequence of activities addressing Scalar Integration in Curvilinear Coordinates. The following activities are included within this sequence:

• Preceding activities:
• Internal Energy of Derivative Machine: This small group activity uses a modified Partial Derivative Machine to measure the internal energy of a nonlinear, one dimensional system while emphasizing integration as an experimentally measurable quantity.
• Curvilinear Coordinates: This lecture introduces students to curvilinear coordinates and highlights the notation difference of $\theta$ and $\phi$ in physics and mathematics.
• Scalar Distance, Area, and Volume Elements: In this small group activity students derive expressions for infinitesimal distances in order to find area and volume elements in cylindrical and spherical coordinates and can be done with Pineapples and Pumpkins to give students a three dimensional object to explore the geometry and construction of a volume element.
• Pineapples and Pumpkins: This activity can be done in small groups or as an instructor led whole class activity where a pineapple (for cylindrical) and/or pumpkin (for spherical) can be cut to demonstrate the geometry of an infinitesimal volume element used in integration.
• Follow-up activities:
• Total Charge: In this small group activity, students calculate the total charge within spherically or cylindrically symmetric volumes by using multivariable integration in various coordinate systems in order to find the total charge contained within the volume due to a specific charge density.

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