Activities
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.
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
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 we demonstrate the answers with coins under a doc cam.
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.
Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.
Problem
- 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.
- On a different cube: 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.
Problem
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 \(x=+d\).
- Sketch the charge distribution.
- Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.
Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
- (4pts) Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
- (4pts) Find a formula for the charge density that creates this electric field.
- (2pts) Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
For each case below, find the total charge.
- (4pts) 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*}
- (4pts) 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*}
The electrostatic potential due to a point charge at the origin is given by: \begin{equation*} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation*}
- (2pts) Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
- (2pts) Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
- (2pts) Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.
(8pts) A charged spiral in the \(x,y\)-plane has 6 turns from the origin out to a maximum radius \(R\) , with \(\phi\) increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the end of the spiral, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the spiral?
One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. 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.
- (2 pts) What are the dimensions of the constants \(\alpha\) and \(k\)?
- (2 pts) By hand, sketch a graph the charge density as a function of \(r\) for \(\alpha > 0\) and \(k>0\) .
- (2 pts) Use step functions to write this charge density as a single function valid everywhere in space.
Problem
Consider the fields at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\).
- Write down an expression for the electrostatic potential \(V(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
- Write down an expression for the electric field \(\vec{E}(\vec{r})\) at a point \(\vec{r}\) due to a point charge located at \(\vec{r}'\). (There is nothing to calculate here.)
- Working in rectangular coordinates, compute the gradient of \(V\).
- Write several sentences comparing your answers to the last two questions.
Consider the volume charge density: \begin{equation*} \rho (x,y,z)=c\,\delta (x-3) \end{equation*}
- (2 pts) Describe in words how this charge is distributed in space.
- (2 pts) What are the dimensions of the constant \(c\)?
Write the equation for the electrostatic potential due to a point charge.
Instructor's Guide
Prerequisite Knowledge
Students will usually have seen the electrostatic potential due to a point charge in their introductory course, but may have trouble recalling it.Whole-Class Conversations
As students try to remember the formula, many will conflate potential, potential energy, force, and electric field. Their answers may have some aspects of each of these. We use this question to get the iconic equation into the students' working memory in preparation for subsequent activities. This question also be used to help student disambiguate these different physical quantities.
Correct answers you're likely to see
\[V=\frac{kq}{r}\]
\[V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\] You may want to discuss which constants to use in which contexts, e.g. \(k\) is short and easy to write, but may be conflated with other uses of \(k\) in a give problem whereas \(\frac{1}{4\pi\epsilon_0}\) assumes you are working in a particular system of units.
Incorrect answers you're likely to see
- Two charges instead of one \[\cancel{V=\frac{kq_{1}q_{2}}{r}}\]
- Distance squared in the denominator \[\cancel{V=\frac{kq}{r^2}}\]
- Vector values \[\cancel{V=\frac{kq\, \hat r}{r}}\]
Possible follow-up questions to help with the disambiguation:
- Relationship between potential and potential energy \(U = qV\)
- Which function is the derivative of the other: \(1/r\) or \(1/r^2\)?
- Which physical quantity (potential or electric field, potential energy or force) is the derivative of the other?
- What is the electrostatic potential conceptually?
- Which function falls off faster: \(1/r\) or \(1/r^2\)?
- What are the dimensions of potential? Units?
- Where is the zero of potential?
Wrap-up
- This could be a good time to refer to the (correct) expression for the potential as an iconic equation, which will need to be further interpreted (”unpacked”) in particular physical situations. This is where the course is going next.
- This SWBQ can also serve to help students learn about recall as a cognitive activity. While parts of the equations that students write may be incorrect, many other parts will be correct. Let the way in which you manage the class discussion model for the students how a professional goes about quickly disambiguating several different choices. And TELL the students that this is what you are doing. Deliberately invoke their metacognition.
- Many students may not know that the electrostatic potential that we are talking about in this activity is the same quantity as what a voltmeter reads, in principle, but not in practice. You may need to talk about how a voltmeter actually works, rather than idealizing it. It helps to have a voltmeter with leads as a prop. Students often want to know about the “ground” lead. We often tie a long string to it (to symbolize making a really long wire) and send the TA out of the room with the string, “headed off to infinity” while discussing the importance of setting the zero of potential. The extra minute or two of humerous byplay gives the importance of the zero of potential a chance to sink in.
We use this small whiteboard question as a transition between The Distance Formula (Star Trek) activity, where students are learning about how to describe (algebraically) the geometric distance between two points, and the Electrostatic Potential Due to a Pair of Charges (with Series) activity, where students are using these results and the superposition principle to find the electrostatic potential due to two point charges.
This activity is the initial activity in the sequence Visualizing Scalar Fields addressing the representations of scalar fields in the context of electrostatics.
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.
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.
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.
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.
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.
In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.
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.
- 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 Two Charged Plates before this activity.
- Students should know that
- objects with like charge repel and opposite charge attract,
- object tend to move toward lower energy configurations
- The potential energy of a charged particle is related to its charge: \(U=qV\)
- The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)