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## Acting Out Current Density: Instructor's Guide

### Main Ideas

Students discuss the concept of current density and how it is measured.

### Students' Task

*Estimated Time: 10 minutes*

To move around the room to mimic specified current densities.

### Prerequisite Knowledge

Students will probably associate the words “charge per time” with the concept of current, but may have no further conceptual understanding. They will probably also be familiar with mass and charge densities as “amount of stuff per length/area/volume.”

We usually do this activity sometime after the Acting Out Charge Densities activity.

### Props/Equipment

- Meter stick
- Magnetic field “meter”

### Activity: Introduction

Prompts: Imagine you are a bunch of charges.

- Make this magnetic field meter fluctuate.
- Keep moving, but in a way that the magnetic field meter does not fluctuate.
- Make the meter read a larger magnitude. What different things can you do to make it read a larger magnitude?
- Make a linear current density. How do we measure linear current density?

### Activity: Student Conversations

**Steady Currents:**Our students often already know that currents generate a magnetic field, so the instructor stands in the middle of the room (often on a table) and asks the students to move in such a way that her “magnetic field meter” doesn't fluctuate. The students quickly realize that their motion must be consistent so that at any point in space, the current doesn't change with time. The instructor then defines this as a*steady*current.**Constant vs. Uniform:**The students often respond by walking in a circle around the instructor. A nice conversation to have with the students is “Does it have to be a circle?”. For charge densities, we use the terms “constant” and “uniform” interchangeably, but with current density, it pays to distinguish between time variation (constant) and spatial variation (uniform).**What does linear mean?:**Students have effectively used the mneumonic “linear mass density is mass per unit length, surface charge density is mass per unit area, …” for both mass and charge densities. They are tempted to use it again here, so linear current density must be current per unit length. As they start to act this out, you will see the look of confusion on their faces. What is*linear*is that it is caused by a linear charge density (times a velocity).**Gates:**Current has dimensions of charge per unit time, and is measured by counting how many charges pass through a gate. If the charge density is a linear charge density (i.e. 1-dimensional), then the gate is a point. If the charge density is a surface density, then the gate is a line segment. If the current density is a volume density, then the gate is a 2-D surface.**Total current:**Total current is the flux of the current density through the gate. Therefore, a linear current density is the “same” as the total current in an idealized 1-dimensional wire.**Current is a flux:**Make sure students get to see what happens if someone goes through a gate in a direction that is not perpendicular to the gate.

### Activity: Wrap-up

**Formalism:** You might want to follow this conceptual activity with a brief lecture/discussion on the formalism and conventions. Write the symbols for linear, surface and volume density on the board ($\Vec{I}$, $\Vec{K}$ and $\Vec{J}$), solicit from the class their appropriate units and dimensions and write the formulas for how the total current is computed from a current density and/or a charge density and velocity. Emphasize that current density is a **flux**.

We follow the discussion in Griffiths' *Introduction to Electrodynamics*, 3rd Ed., pp. 208-214). See also

- GVC § Current

### Extensions

This activity works particularly well if the students have already done the activity

This activity is included within a sequence of activities addressing Ampere’s law. The following activities are additional activities which are included within this sequence.

- Preceding activity:
- Gauss's Law: A small group activity which students actively make symmetry arguments by
*Proof by Contradiction*to calculate the electric field due to a highly symmetric charge distribution using Gauss's law.

- Follow-up activities:
- Ampere's Law and Symmetry Argument Lecture: This lecture serves as an introduction to the Ampere's Law activity by refreshing students about the flux of a current density from Acting Out Current Density, introducing Ampere's law, and discussing
*Proof by Contradiction*in detail. - Ampere's Law: A small group activity which uses similar skills as those developed in the Gauss's Law activity by extending symmetry arguments using
*Proof by Contradiction*in Ampere's law with highly symmetric current densities.