## Activity: Acting Out Current Density

Static Fields 2022 (5 years)
Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear $\vec{I}$, surface $\vec{K}$, and volume $\vec{J}$ current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
• This activity is used in the following sequences
What students learn
• Conceptual/geometric understanding of steady linear, surface and volume currents,
• The distinction between uniform and non-uniform current densities,
• Current density increases if the charges increase, the charges are closer together, or the charges move faster.
• Current density is charge density time the velocity of the charged particles,
• Current density is a flux. It is proportional to the rate at which charges pass through a gate.

## Instructor's Guide

### Introduction

Students move around the room acting out various prompts from the instructor regarding current densities of different dimensions.

Tell the students that they each represent a point charge. Then initiate a conversation with the whole class by asking the prompts listed in "Student Conversations," below. Be flexible about the order of the prompts, responding to the ideas brought up by the students.

We usually hold a voltmeter and pretend that it is set to measure "magnetic field". Don't forget to mention that this use of a voltmeter is metaphorical and unrealistic.

Note: It helps if the instructor stands on a chair or table so they are high enough to hold a meter stick above the "current" that the students make. The meter stick represents the "gate" used to measure the total current.

We usually do this activity sometime after the Activity: Acting Out Charge Densities. The embodied understanding here builds on the previous activity.

### Student Conversations

Sample Prompts: "Stand up. Imagine you are each a point charge."

• "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?"

Weave the following discussions in amongst the prompts at appropriate times:

• 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 their motion as a steady current.
• What can change?: 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?."
• Gates: Current has dimensions of charge per unit time and "charge per time" may be the only concept of current that some students can state. Demonstrate that these words mean counting how many charges pass through a "gate".
• What does linear current density 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 the mneumonic 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 the current is caused by a linear charge density (times a velocity).
• Dimensions of gates: 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.

### 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 also write the formulas \begin{align*} \vec{I}&=\lambda\, \vec{v}\\ \vec{K}&=\sigma\, \vec{v}\\ \vec{J}&=\rho\, \vec{v} \end{align*} that show current density is an appropriate charge density times velocity.

If the students have already encountered the concept of flux, emphasize that TOTAL current is a flux through a gate. \begin{align*} I_{\hbox{total}} &=\vec{I}\cdot\hat{n}\\ I_{\hbox{total}} &=\int\vec{K}\cdot\hat{n}\,\vert d\vec{r}\vert\\ I_{\hbox{total}} &=\int\vec{J}\cdot\hat{n}\, dA \end{align*} where $\hat{n}$ is perpendicular to the gate. Associated reading: see Griffiths' Introduction to Electrodynamics, 3rd Ed., pp. 208-214 and Current

• group Magnetic Vector Potential Due to a Spinning Charged Ring

group Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the superposition principle $\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}$ to find an integral expression for the magnetic vector potential, $\vec{A}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{A}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• assignment Total Current, Square Cross-Section

assignment Homework

##### Total Current, Square Cross-Section

Integration Sequence

Static Fields 2022 (5 years)
1. Current $I$ flows down a wire with square cross-section. The length of the square side is $L$. If the current is uniformly distributed over the entire area, find the current density .
2. If the current is uniformly distributed over the outer surface only, find the current density .
• assignment Total Current, Circular Cross Section

assignment Homework

##### Total Current, Circular Cross Section

Integration Sequence

Static Fields 2022 (4 years)

A current $I$ flows down a cylindrical wire of radius $R$.

1. If it is uniformly distributed over the surface, give a formula for the surface current density $\vec K$.
2. If it is distributed in such a way that the volume current density, $|\vec J|$, is inversely proportional to the distance from the axis, give a formula for $\vec J$.

• assignment Magnetic Field and Current

assignment Homework

##### Magnetic Field and Current
Static Fields 2022 (3 years) Consider the magnetic field $\vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases}$
1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
2. Find a formula for the current density that creates this magnetic field.
3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
• assignment Current from a Spinning Cylinder

assignment Homework

##### Current from a Spinning Cylinder
A solid cylinder with radius $R$ and height $H$ has its base on the $x,y$-plane and is symmetric around the $z$-axis. There is a fixed volume charge density on the cylinder $\rho=\alpha z$. If the cylinder is spinning with period $T$:
1. Find the volume current density.
2. Find the total current.
• accessibility_new Acting Out Charge Densities

accessibility_new Kinesthetic

10 min.

##### Acting Out Charge Densities
Static Fields 2022 (5 years)

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear $\lambda$, surface $\sigma$, and volume $\rho$ charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.
• assignment Current in a Wire

assignment Homework

##### Current in a Wire
Static Fields 2022 (3 years) The current density in a cylindrical wire of radius $R$ is given by $\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}$. Find the total current in the wire.
• format_list_numbered Ring Cycle Sequence

format_list_numbered Sequence

##### Ring Cycle Sequence
Students calculate electrostatic fields ($V$, $\vec{E}$) and magnetostatic fields ($\vec{A}$, $\vec{B}$) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.
• group Magnetic Field Due to a Spinning Ring of Charge

group Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{B}(\vec{r})$ either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

• face Guide to Professional Typography in Physics

face Lecture

5 min.

##### Guide to Professional Typography in Physics
Contemporary Challenges 2022 (4 years)

This is really a handout, which gives students guidelines on how to type up physics content.

Author Information
Corinne Manogue
Keywords
Steady current current density magnetic field idealization
Learning Outcomes