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.
1. << Flux through a Cone | Integration Sequence | Total Current, Circular Cross Section >>
2. << Electric Field Due to a Ring of Charge | Ring Cycle Sequence | Magnetic Vector Potential Due to a Spinning Charged Ring >>
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.
Sample Prompts: "Stand up. Imagine you are each a point charge."
Weave the following discussions in amongst the prompts at appropriate times:
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 Small Group Activity
30 min.
compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry
Students work in small groups 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 Homework
accessibility_new Kinesthetic
10 min.
density charge density mass density linear density uniform idealization
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 Homework
A current \(I\) flows down a cylindrical wire of radius \(R\).
assignment Homework
assignment Homework
assignment Homework
format_list_numbered Sequence
group Small Group Activity
30 min.
magnetic fields current Biot-Savart law vector field symmetry
Students work in small groups 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 Lecture
5 min.