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.
1. << Linear Quadrupole (w/ series) | Power Series Sequence (E&M) | Electric Field Due to a Ring of Charge >>
2. << Electrostatic Potential Due to a Point Charge | Warm-Up |
3. << Acting Out Charge Densities | Ring Cycle Sequence | Electric Field Due to a Ring of Charge >>
The Electrostatic Potential Due to a Ring of Charge
Use the superposition principle for the electrostatic potential due to a continuous charge distribution: \begin{align} V(\vec{r})=\frac{1}{4\pi \epsilon_0} \int \frac{\rho'(\vec{r}^{\,\prime})}{\left| \vec{r}-\vec{r}'\right|}\, d\tau', \end{align} to find the electrostatic potential everywhere in space due to a uniformly charged ring with radius \(R\) and total charge \(Q\).
Check with a teaching team member before moving on to subsequent parts below.
- Evaluate your expression for the special case of the potential on the \(z\)-axis.
- Evaluate your expression for the special case of the potential on the \(x\)-axis.
- Find a series expansion for the electrostatic potential in these special regions:
- 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.
Part I - Finding the potential everywhere in space
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 \(Q\). Find a formula for the electrostatic potential \(V\) due to this ring that is valid everywhere in space".
Part II (Optional) - Power series expansion along an axis
With the charged ring in the \(x,y\)-plane, student groups are asked to make the power series expansion for either near or far from the plane on the \(z\) axis or near or far from the \(z\) axis in the \(x,y\)-plane. The instructor may choose to have the whole class do one particular case or have different groups do different cases, in a Compare and Contrast strategy (Compare and Contrast Activities).
Part I - Finding the potential everywhere in space
Part II (Optional) - Power series expansion along an axis
group Small Group Activity
30 min.
coulomb's law electric field charge ring symmetry integral power series superposition
Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.
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.
magnetic fields current Biot-Savart law vector field symmetry
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.
group Small Group Activity
30 min.
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?
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
Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r'}\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert} \] Electrostatic potentials satisfy the superposition principle.
Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.
format_list_numbered Sequence
assignment_ind Small White Board Question
10 min.
assignment Homework
In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences: