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Activities

Kinesthetic

10 min.

Acting Out Current Density
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.

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.

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 \(\vec J\).

  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Integration Sequence sequence(s)
  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 .
  • Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Integration Sequence, Flux Sequence sequence(s)
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. (2pts) Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
  2. (4pts) Find a formula for the current density that creates this magnetic field.
  3. (2pts) Interpret your formula for the current density, i.e. explain briefly in words where the current is.
  • Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s)
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.
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.
  • Found in: AIMS Maxwell, Static Fields, Problem-Solving course(s)

Small White Board Question

30 min.

Magnetic Moment & Stern-Gerlach Experiments
Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

Lecture

10 min.

Warm-Up Powerpoint
The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU. Make sure to update this powerpoint yearly to reflect current course offerings and sequencing. It was partially, but not completely edited in fall 2022.
  • Found in: Warm-Up sequence(s)