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.
1. << Series Notation 2 | Power Series Sequence (E&M) |
2. << Magnetic Field Due to a Spinning Ring of Charge | E&M Ring Cycle Sequence |
The Magnetic Vector Potential Due to a Spinning Ring of Charge
- Use the superposition principle for the magnetic vector potential due to a continuous current distribution: \begin{align} \vec{A}(\vec{r})=\frac{\mu_0}{4\pi} \int \frac{\vec{J}'(\vec{r}^{\,\prime})}{\left| \vec{r}-\vec{r}'\right|}\, d\tau', \end{align} to find the magnetic vector potential everywhere in space due to a spinning charged ring with radius \(R\), total charge \(Q\), and period \(T\).
- Evaluate your expression for the special case that \(\vec{r}\) is on the \(z\)-axis.
- Evaluate your expression for the special case that \(\vec{r}\) is on the \(x\)-axis.
- Find a series expansion for the electrostatic potential at these special locations:
- Near the center of the ring, in the plane of the ring;
- Near the center of the ring, on the axis of the ring;
- Far from the ring on the axis of symmetry;
- Far from the ring, in the plane of the ring.
Students should be assigned to work in small groups given the following instructions using the visual of a hula hoop or other large ring:
Prompt: "This is a ring with radius \(R\) and total charge \(Q\) and spinning with period \(T\). Find a formula for the magnetic vector potential \(\vec{A}\) due to this ring that is valid everywhere in space".
This activity is part of a sequence (the E&M Ring Cycle Sequence) of four electrostatics activities involving a ring of charge: \(V\), \(\vec{E}\), \(\vec{A}\), \(\vec{B}\). They are arranged so that the mathematical complexity of the problems increases in a natural way. If you are doing this activity as a standalone, please see the Student Conversations section of the previous activities (Electrostatic Potential Due to a Ring of Charge, Electric Field Due to a Ring of Charge) for further advice.
Part I - Finding the potential everywhere in space
The new idea in the magnetic vector potential case is to find the linear current density (current) in the ring. Many students will have learned that current is “charge per time” or the derivative of charge with respect to time. Neither of these resources about current will be helpful to them here. They will need to know that current density is charge density times velocity: \begin{align} \vec{I}&=\lambda \vec{v}\\ &=\frac{Q}{2\pi R}\, \frac{2 \pi R}{T} \hat{\phi} \end{align} Be watchful. Many students will get the correct answer on dimensional grounds, but will not be able to justify their answer in a way that will extend to other problems.
Part II - Evaluate on the \(z\)-axis and on the \(x\)-axis
(See solution.)
Part III (Optional) - Series expansions
If you are doing this activity as a standalone, please see the Wrap-Up section of the previous activities (Electrostatic Potential Due to a Ring of Charge, Electric Field Due to a Ring of Charge) for further advice.