assignment Homework

Volume Charge Density, Version 2
charge density delta function Static Fields 2022 (4 years)

You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge \(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).

  1. Sketch the charge distribution.
  2. Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.

assignment Homework

Central Force
Central Forces 2021

Which of the following forces can be central forces? which cannot?

  1. The force on a test mass \(m\) in a gravitational field \(\vec{g~}\), i.e. \(m\vec g\)
  2. The force on a test charge \(q\) in an electric field \(\vec E\), i.e. \(q\vec E\)
  3. The force on a test charge \(q\) moving at velocity \(\vec{v~}\) in a magnetic field \(\vec B\), i.e. \(q\vec v \times \vec B\)

assignment Homework

Spherical Shell Step Functions
step function charge density Static Fields 2022 (4 years)

One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. If you need to review this, see the following link in the math-physics book: https://books.physics.oregonstate.edu/GMM/step.html

Consider a spherical shell with charge density \(\rho (\vec{r})=\alpha3e^{(k r)^3}\) between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else. Use step functions to write this charge density as a single function valid everywhere in space.

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

Volume Charge Density
Static Fields 2022 (4 years)

Sketch the volume charge density: \begin{equation} \rho (x,y,z)=c\,\delta (x-3) \end{equation}

assignment Homework

Gauss's Law for a Rod inside a Cube
Static Fields 2022 (3 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(x,y\)-plane. The charge density \(\lambda_0\) is constant. Find the total flux of the electric field through a closed cubical surface with sides of length \(3L\) centered at the origin.

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?

assignment Homework

Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2022 (4 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}

assignment Homework

Electric Field from a Rod
Static Fields 2022 (4 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(xy\)-plane. The charge density \(\lambda\) is constant. Find the electric field at the point \((0,0,2L)\).

assignment Homework

Cube Charge
charge density

Integration Sequence

Static Fields 2022 (4 years)
  1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
  2. In a new physical situation: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.

computer Computer Simulation

30 min.

Visualizing Flux through a Cube
Static Fields 2022 (4 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.

group Small Group Activity

30 min.

Total Charge
Static Fields 2022 (4 years)

charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

Integration Sequence

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

accessibility_new Kinesthetic

10 min.

Acting Out Charge Densities
Static Fields 2022 (4 years)

density charge density mass density linear density uniform idealization

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.

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/ 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.

group Small Group Activity

30 min.

Electric Field of Two Charged Plates
  • Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity “Electric Potential of a Parallel Plate Capacitor” before this activity.
  • Students should know that
    1. objects with like charge repel and opposite charge attract,
    2. object tend to move toward lower energy configurations
    3. The potential energy of a charged particle is related to its charge: \(U=qV\)
    4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

assignment Homework

Helix

Integration Sequence

Static Fields 2022 (4 years)

A helix with 17 turns has height \(H\) and radius \(R\). Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix. At the bottom of the helix the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the top of the helix, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the helix?

assignment Homework

Differential Form of Gauss's Law
Static Fields 2022 (4 years)

For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

accessibility_new Kinesthetic

10 min.

Acting Out Current Density
Static Fields 2022 (4 years)

Steady current current density magnetic field idealization

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

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.