Activity: Magnetic Vector Potential Due to a Spinning Charged Ring

Static Fields 2024 (9 years)

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.

What students learn
  • to perform a magnetic vector potential calculation using the superposition principle;
  • to decide which form of the superposition principle to use, depending on the dimensions of the current density;
  • how to find current from total charge \(Q\), period \(T\), and the geometry of the problem, radius \(R\);
  • to write the distance formula \(\vec{r}-\vec{r'}\) in both the numerator and denominator of the superposition principle in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
The Magnetic Vector Potential Due to a Spinning Ring of Charge
  1. 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 \(T\).
  2. Evaluate your expression for the special case that \(\vec{r}\) is on the \(z\)-axis.
  3. Evaluate your expression for the special case that \(\vec{r}\) is on the \(x\)-axis.
  4. Find a series expansion for the electrostatic potential at these special locations:
    1. Near the center of the ring, in the plane of the ring;
    2. Near the center of the ring, on the axis of the ring;
    3. Far from the ring on the axis of symmetry;
    4. Far from the ring, in the plane of the ring.
Author Information
Corinne Manogue, Leonard Cerny
Keywords
compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry
Learning Outcomes