Magnetic Vector Potential Due to a Spinning Charged Ring

Student handout: Magnetic Vector Potential Due to a Spinning Charged Ring

Static Fields 2023 (6 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.

Small Group Activity schedule 30 min. build whiteboards/markers/erasers, hula hoop, coordinate axes in the ceiling, imaginary magnetic vector potential meter, prompt, graph of the solution description Student handout (PDF)
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compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

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

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 \(T\).
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:
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.

Magnetic Field Due to a Spinning Ring of Charge

Small Group Activity

30 min.

Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2023 (7 years)
magnetic fields current Biot-Savart law vector field symmetry

Power Series Sequence (E&M)

Ring Cycle Sequence

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.
Electrostatic Potential Due to a Ring of Charge

Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2023 (8 years)
electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula

Power Series Sequence (E&M)

Warm-Up

Ring Cycle Sequence

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.
Electric Field Due to a Ring of Charge

Small Group Activity

30 min.

Electric Field Due to a Ring of Charge
Static Fields 2023 (8 years)
coulomb's law electric field charge ring symmetry integral power series superposition

Power Series Sequence (E&M)

Ring Cycle Sequence

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.
Total Charge
Homework

Total Charge
charge density curvilinear coordinates
Integration Sequence
Static Fields 2023 (6 years)
For each case below, find the total charge.
1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}
Mass of a Slab
Homework

Mass of a Slab
Static Fields 2023 (6 years)
Determine the total mass of each of the slabs below.
1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
5. What are the dimensions of \(A\)?
6. Write several sentences comparing your answers to the different cases above.
Unit Learning Outcomes: Quantum Mechanics on a Ring

Lecture

5 min.

Unit Learning Outcomes: Quantum Mechanics on a Ring
Central Forces 2023
Electric field for a waffle cone of charge

Computational Activity

120 min.

Electric field for a waffle cone of charge
Computational Physics Lab II 2022
electric field cone
Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
Superposition States for a Particle on a Ring

Small Group Activity

30 min.

Superposition States for a Particle on a Ring

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

Quantum Ring Sequence
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.
Quantum Ring Sequence

Sequence

Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.
Time Dependence for a Quantum Particle on a Ring Part 1

Small Group Activity

30 min.

Time Dependence for a Quantum Particle on a Ring Part 1
Theoretical Mechanics (6 years)
central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

Quantum Ring Sequence
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.

Author Information

Corinne Manogue, Leonard Cerny

Learning Outcomes