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. << Magnetic Vector Potential Due to a Spinning Charged Ring | Power Series Sequence (E&M) | Series Notation 1 >>
2. << Magnetic Vector Potential Due to a Spinning Charged Ring | 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 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.
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.
group Small Group Activity
30 min.
electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula
Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) 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.
group Small Group Activity
60 min.
electrostatic potential multipole charge symmetry scalar field superposition coulomb's Law
Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.assignment Homework
accessibility_new Kinesthetic
30 min.
distance formula coordinate systems dot product vector addition
A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. \[\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}\]assignment Homework
Which pairs of events (if any) are simultaneous in the unprimed frame?
Which pairs of events (if any) are simultaneous in the primed frame?
Which pairs of events (if any) are colocated in the unprimed frame?
Which pairs of events (if any) are colocated in the primed frame?
Which event occurs first in the unprimed frame?
Which event occurs first in the primed frame?
group Small Group Activity
5 min.
Special Relativity Spacetime Diagrams Simultaneity Colocation
Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.assignment Homework
Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.
A series of charges arranged in this way is called a linear quadrupole. Why?
assignment Homework
Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.
You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}