format_list_numbered Sequence

Ring Cycle Sequence
Students calculate electrostatic fields (\(V\), \(\vec{E}\)) and magnetostatic fields (\(\vec{A}\), \(\vec{B}\)) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.

assignment Homework

Total Current, Circular Cross Section

Integration Sequence

Static Fields 2022 (3 years)

A current \(I\) flows down a cylindrical wire of radius \(R\).

  1. If it is uniformly distributed over the surface, give a formula for the surface current density \(\vec K\).
  2. If it is distributed in such a way that the volume current density, \(|\vec J|\), is inversely proportional to the distance from the axis, give a formula for \(\vec J\).

accessibility_new Kinesthetic

30 min.

The Distance Formula (Star Trek)
Static Fields 2022 (4 years)

distance formula coordinate systems dot product vector addition

Ring Cycle Sequence

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

Distance Formula in Curvilinear Coordinates

Ring Cycle Sequence

Static Fields 2022 (4 years)

The distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

  1. Find the distance \(\left\vert\vec r -\vec r\,{}'\right\vert\) between the point \(\vec r\) and the point \(\vec r'\) in rectangular coordinates.
  2. Show that this same distance written in cylindrical coordinates is: \begin{equation} \left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2} \end{equation}
  3. Show that this same distance written in spherical coordinates is: \begin{equation} \left\vert\vec r -\vec r\,{}'\right\vert =\sqrt{r'^2+r\,{}^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}') +\cos\theta\cos\theta\,{}'\right]} \end{equation}
  4. Now assume that \(\vec r\,{}'\) and \(\vec r\) are in the \(x\)-\(y\) plane. Simplify the previous two formulas.

group Small Group Activity

30 min.

Electrostatic Potential Due to a Pair of Charges (without Series)
Static Fields 2022 (3 years) 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). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.

assignment Homework

Linear Quadrupole (w/o series)
Static Fields 2022 (3 years) Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
  1. Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.

  2. A series of charges arranged in this way is called a linear quadrupole. Why?

group Small Group Activity

60 min.

Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (4 years)

electrostatic potential multipole charge symmetry scalar field superposition coulomb's Law

Power Series Sequence (E&M)

Ring Cycle Sequence

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.

group Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (6 years)

electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three 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

Circle Vector, Version 2
Static Fields 2022 (4 years)

Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.

  1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
  2. Make a sketch of the graph \begin{equation} \vert \vec{r} - \vec{a} \vert = 2 \end{equation}

    for each of the following values of \(\vec a\): \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

  3. Derive a more familiar equation equivalent to \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation} for arbitrary \(\vec a\), by expanding \(\vec r\) and \(\vec a\) in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
  4. Write a brief description of the geometric meaning of the equation \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation}

group Small Group Activity

30 min.

Electric Field Due to a Ring of Charge
Static Fields 2022 (6 years)

coulomb's law electric field charge ring symmetry integral power series superposition

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three 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.

assignment Homework

Linear Quadrupole (w/ series)

Power Series Sequence (E&M)

Static Fields 2022 (4 years)

Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

  1. 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.
  2. 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.

assignment Homework

Scattering
Central Forces 2022 (2 years)

Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

  1. What is the initial angular momentum of the system?
  2. What is the initial total energy of the system?
  3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
  4. Sketch the effective potential.
  5. What is the angular momentum at \(r_{\rm{min}}\)?
  6. What is the total energy of the system at \(r_{\rm{min}}\)?
  7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
  8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
  9. What is the value of the effective potential at \(r_{\rm{min}}\)?
  10. For what values of the initial total energy are there bound orbits?
  11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

assignment Homework

Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

  1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
  2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

group Small Group Activity

30 min.

Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (5 years)

magnetic fields current Biot-Savart law vector field symmetry

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three 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.

Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (4 years)

compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three 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.