Activity: Acting Out Current Density

AIMS Maxwell AIMS 21 Static Fields Winter 2021
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.
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.
  • 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
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

    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.

  • accessibility_new Acting Out Charge Densities

    accessibility_new Kinesthetic

    10 min.

    Acting Out Charge Densities
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    density charge density mass density linear density uniform idealization

    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 Total Current, Square Cross-Section

    assignment Homework

    Total Current, Square Cross-Section

    Integration Sequence

    AIMS Maxwell AIMS 21 Static Fields Winter 2021
    1. Current \(I\) flows down a wire (length \(L\)) with square cross-section (side \(a\)). If it is uniformly distributed over the entire area, what is the magnitude of the volume current density \(\vec{J}\)?
    2. If the current is uniformly distributed over the outer surface only, what is the magnitude of the surface current density \(\vec{K}\)?
  • assignment Total Current, Circular Cross Section

    assignment Homework

    Total Current, Circular Cross Section

    Integration Sequence

    AIMS Maxwell AIMS 21

    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
    AIMS Maxwell AIMS 21 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.
  • assignment Current in a Wire

    assignment Homework

    Current in a Wire
    AIMS Maxwell AIMS 21 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
    AIMS Maxwell AIMS 21 Static Fields Winter 2021

    magnetic fields current Biot-Savart law vector field symmetry

    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.

  • assignment Ring Function

    assignment Homework

    Ring Function
    Central Forces Spring 2021 Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
    1. Find \(N\).

    2. Plot this wave function.
    3. Plot the probability density.
    4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
    5. What is the expectation value of \(L_z\) in this state?

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


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