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.
1. << Electrostatic Potential Due to a Ring of Charge | Power Series Sequence (E&M) | Magnetic Vector Potential Due to a Spinning Charged Ring >>
2. << Electrostatic Potential Due to a Ring of Charge | Ring Cycle Sequence | Acting Out Current Density >>
The Electrostatic Field Due to a Ring of Charge
- 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 the electric field everywhere in space due to a charged ring with radius \(R\) and total charge \(Q\).
- 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 electric field 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 groupsand 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 electric field \(\vec{E}\) 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 activity (Electrostatic Potential Due to a Ring of Charge) for further advice.
Part I - Finding the electric field everywhere in space
The new idea in the electric field case is that the numerator is a vector. The basis vectors in cylindrical or spherical coordinates differ from point to point in space. Therefore, you CANNOT subtract two vectors that "live" at different points if they are expanded in curvilinear coordinate basis vectors. The numerator in this case must be expanded in rectangular basis vectors (so you can subtract) and components written in curvilinear coordinates (so that you can integrate)
\[\vec{r}-\vec{r}^{\prime}=(s\cos{\phi}-R\cos{\phi^{\prime}})\hat{x}+(s\sin{\phi}-R\sin{\phi^{\prime}})\hat{y}+(z)\hat{z} \]
Part II (Optional) - Power series expansion along an axis
This part will go much the same as for the potential case.
If you are doing this activity as a standalone, please see the Wrap-Up section of the previous activity (Electrostatic Potential Due to a Ring of Charge) for further advice.
New wrap-up content for this activity:
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.
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.
assignment Homework
Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition
Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.assignment Homework
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.group Small Group Activity
30 min.
group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy
Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.