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.
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 Field Due to a Spinning Ring of Charge
- 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 the magnetic field 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 and 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 field \(\vec{B}\) 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, Magnetic Vector Potential Due to a Spinning Charged Ring) for further advice.
Part I - Finding the magnetic field everywhere in space
The new idea in the magnetic field case is the cross product in the numerator. Many students will find this cross product using rectangular basis vectors, which is correct. But they may NOT realize that curvilinear basis vectors AT A SINGLE POINT are also orthonormal and it is perfectly correct to find the cross product using them IF the curvilinear basis vectors are both primed OR both unprimed, but not otherwise.
Part II (Optional) - Power series expansion along an axis
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, Magnetic Vector Potential Due to a Spinning Charged Ring) for further advice.