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.

Calculating Total Charge

Each group will be given one of the charge distributions given below: (\(\alpha\) and \(k\) are constants with dimensions appropriate for the specific example.)

• Spherical Symmetery

  1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, r^{3}\)

  2. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}\)

  3. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{r^{2}}\, e^{(kr)}\)

• Cylindrical Symmetry

  1. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, s^{3}\)

  2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}\)

  3. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \(\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}\)

For your group's case, answer the following questions:

1. Find the total charge. (If the total charge is infinite, decide what you should calculate instead to provide a meaningful answer.)
2. Find the dimensions of the constants \(\alpha\) and \(k\).

Instructor's Guide

Introduction

We usually start with a mini-lecture reminder that total charge is calculated by integrating over the charge density by chopping up the charge density, multiplying by the appropriate geometric differential (length, area, or volume element), and adding up the contribution from each of the pieces. Chop, Multiply, Add is a mantra that we want students to use whenever they are doing integration in a physical context.

The students should already know formulas for the volume elements in cylindrical and spherical coordinates. We recommend Scalar Surface and Volume Elements as a prerequisite.

We start the activity with the formulas \(Q=\int\rho(\vec{r}')d\tau'\), \(Q=\int\sigma(\vec{r}')dA'\), and \(Q=\int\lambda(\vec{r}')ds'\) written on the board. We emphasize that choosing the appropriate formula by looking at the geometry of the problem they are doing, is part of the task.

Each student group is assigned a particular charge density that varies in space and asked to calculate the total charge. This activity is an example of https://paradigms.oregonstate.edu/whitepaper/compare-and-contrast-activity.

Student Conversations

This activity helps students practice the mechanics of making total charge calculations.

• Order of Integration When doing multiple integrals, students rarely think about the geometric interpretation of the order of integration. If they do the \(r\) integral first, then they are integrating along a radial line. What about \(\theta\) and \(\phi\). If this topic does not come up in the small groups, it makes a rich discussion in the wrap-up.
• Limits of Integration some students need some practice determining the limits of the integrals. This issue becomes especially important for the groups working with a cylinder - the handout does not give the students a height of the cylinder. There are two acceptable resolutions to this situation. Students can “name the thing they don't know” and leave the height as a parameter of the problem. Students can also give the answer as the total charge per unit length. We usually talk the groups through both of these options.
• Dimensions Students have some trouble determining the dimensions of constants. Making students talk through their reasoning is an excellent exercise. In particular, they should know that the argument of the exponential function (indeed, the argument of any special fuction other than the logarithm) must be dimensionless.
• Integration Some students need a refresher in integrating exponentials and making \(u\)-substitutions.

Wrap-up

You might ask two groups to present their solutions, one spherical and one cylindrical so that everyone can see an example of both. Examples (b) and (f) are nice illustrative examples.

• The superposition principle for the electrostatic potential;
• How to calculate the distance formula \(\frac{1}{|\vec{r} - \vec{r}'|}\) for a simple specific geometric situation;
• How to calculate the first few terms of a (binomial) power series expansion by factoring out the dimensionful quantity which is large;
• How the symmetries of a physical situation are reflected in the symmetries of the power series expansion.

Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

A pretzel is to be dipped in chocolate. The pretzel is in the shape of a quarter circle, consisting of a straight segment from the origin to the point (2,0), a circular arc from there to (0,2), followed by a straight segment back to the origin; all distances are in centimeters. The (linear) density of chocolate on the pretzel is given by \(\lambda = 3(x^ 2 + y^2 )\) in grams per centimeter. Find the total amount of chocolate on the pretzel.

Main ideas

• Calculating (scalar) line integrals.
• Use what you know!

Prerequisites

• Familiarity with \(d\boldsymbol{\vec{r}}\).
• Familiarity with “Use what you know” strategy.

Warmup

It is not necessary to explicitly introduce scalar line integrals, before this lab; figuring out that the (scalar) line element must be \(|d\boldsymbol{\vec{r}}|\) can be made part of the activity (if time permits).

Props

• whiteboards and pens
• “linear” chocolate covered candy (e.g. Pocky)

Wrapup

Emphasize that students must express each integrand in terms of a single variable prior to integration.

Emphasize that each integral must be positive!

Discuss several different ways of doing this problem (see below).

Details

In the Classroom

• Make sure the shape of the pretzel is clear! It might be worth drawing it on the board.
• Some students will work geometrically, determining \(ds\) on each piece by inspection. This is fine, but encourage such students to try using \(d\vec{r}\) afterwards.
• Polar coordinates are natural for all three parts of this problem, not just the circular arc.
• Many students will think that the integral “down” the \(y\)-axis should be negative. They will argue that \(ds=dy\), but the limits are from \(2\) to \(0\). The resolution is that \(ds = |dy\,\boldsymbol{\hat x}|=|dy|=-dy\) when integrating in this direction.
• Unlike work or circulation, the amount of chocolate does not depend on which way one integrates, so there is in fact no need to integrate “down” the \(y\)-axis at all.
• Some students may argue that \(d\boldsymbol{\vec{r}}=\boldsymbol{\hat T}\,ds\Longrightarrow ds=d\boldsymbol{\vec{r}}\cdot\boldsymbol{\hat T}\), and use this to get the signs right. This is fine if it comes up, but the unit tangent vector \(\boldsymbol{\hat T}\) is not a fundamental part of our approach.
• There is of course a symmetry argument which says that the two “legs” along the axes must have the same amount of chocolate --- although some students will put a minus sign into this argument!

Subsidiary ideas

• \(ds=|d\boldsymbol{\vec{r}}|\)

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.

Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

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 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 are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

Gauss's Law: \[ \oint \vec{E}\cdot \hat{n}\, dA = {1\over\epsilon_0}\, Q_{\hbox{enc}} \]

Ampère's Law:

\[ \oint \vec{B}\cdot d\vec{r} = \mu_0 \, I_{\hbox{enc}} \]

Potentials: \begin{eqnarray*} \vec{E}&=&-\vec{\nabla} V\\ \vec{B}&=&\vec{\nabla}\times\vec{A} \end{eqnarray*}

Maxwell's Equations: \begin{eqnarray*} \vec{\nabla}\cdot\vec{E} &=& \frac{\rho}{\epsilon_0}\\ \vec{\nabla}\cdot\vec{B} &=& 0\\ \vec{\nabla}\times\vec{E} &=& 0\\ \vec{\nabla}\times\vec{B} &=& {\mu_0}\, \vec{J} \end{eqnarray*}

Superposition Laws: \begin{eqnarray*} V(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{E}(\vec{r}) &=& \frac{1}{4\pi\epsilon_0} \int{\rho(\vec{r}')(\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ \vec{A}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert}\\ \vec{B}(\vec{r}) &=& \frac{\mu_0}{4\pi} \int{\vec{J}(\vec{r}')\times (\vec{r}-\vec{r}')\, d\tau'\over \vert \vec{r}-\vec{r}'\vert^3}\\ V(B)-V(A)&=&-\int_A^B \vec{E}\cdot d\vec{r} \end{eqnarray*}

Position Vectors \begin{align*} \vec{r} &= x \hat{x} + y\hat{y} + z\hat{z}\\ &= s \hat{s} + z\hat{z}\\ &= r\hat{r} \end{align*}

The distance between two position vectors

1. In cylindrical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{s^2+s^{\prime\, 2}-2s\, s^{\prime}\cos(\phi- \phi^{\prime}) +(z-z^{\prime})^2}\]
2. In spherical coordinates: \[\left\vert\vec r -\vec r^{\prime}\right\vert =\sqrt{r^2+r^{\prime\, 2}-2r\, r^{\prime}\left[ \sin\theta\sin\theta^{\prime}\cos(\phi-\phi^{\prime}) +\cos\theta\cos\theta^{\prime}\right]}\]

Rectangular Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial x}\,\hat{x} + \frac{\partial f}{\partial y}\,\hat{y} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)\hat{x} + \left(\frac{\partial F_x}{\partial z} -\frac{\partial F_z}{\partial x}\right)\hat{y} + \left(\frac{\partial F_y}{\partial x} -\frac{\partial F_x}{\partial y}\right)\hat{z} \end{eqnarray*}

Cylindrical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial s}\,\hat{s} + \frac{1}{s}\frac{\partial f}{\partial \phi}\,\hat{\phi} + \frac{\partial f}{\partial z}\,\hat{z} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{s}\frac{\partial}{\partial s}\left({s}F_{s}\right) + \frac{1}{s}\frac{\partial F_\phi}{\partial \phi} + \frac{\partial F_z}{\partial z} \\ \vec{\nabla}\times\vec{F} &=& \left( \frac{1}{s}\frac{\partial F_z}{\partial \phi} - \frac{\partial F_\phi}{\partial z} \right) \hat{s} + \left(\frac{\partial F_s}{\partial z}-\frac{\partial F_z}{\partial s}\right) \hat{\phi} + \frac{1}{s} \left( \frac{\partial}{\partial s}\left({s}F_{\phi}\right) - \frac{\partial F_s}{\partial \phi} \right) \hat{z} \end{eqnarray*}

Spherical Coordinates: \begin{eqnarray*} \vec{\nabla} f &=& \frac{\partial f}{\partial r}\,\hat{r} + \frac{1}{r}\frac{\partial f}{\partial \theta}\,\hat{\theta} + \frac{1}{r\sin\theta}\frac{\partial f}{\partial \phi}\,\hat{\phi} \\ \vec{\nabla}\cdot\vec{F} &=& \frac{1}{r^2}\frac{\partial}{\partial r}\left({r^2}F_{r}\right) + \frac{1}{r\sin\theta}\frac{\partial}{\partial \theta}\left({\sin\theta}F_{\theta}\right) + \frac{1}{r\sin\theta}\frac{\partial F_\phi}{\partial \phi} \\ \vec{\nabla}\times\vec{F} &=& \frac{1}{r\sin\theta} \left( \frac{\partial}{\partial \theta} \left({\sin\theta}F_{\phi}\right) - \frac{\partial F_\theta}{\partial \phi} \right) \hat{r} + \frac{1}{r} \left( \frac{1}{\sin\theta} \frac{\partial F_r}{\partial \phi} - \frac{\partial}{\partial r}\left({r}F_{\phi}\right) \right) \hat{\theta} \\ && \quad + \frac{1}{r} \left( \frac{\partial}{\partial r}\left({r}F_{\theta}\right) - \frac{\partial F_r}{\partial \theta} \right) \hat{\phi} \end{eqnarray*}

Lorentz Force Law:

\[\vec{F}=q_{\hbox{test}}\left(\vec{E}+\vec{v}\times\vec{B}\right)\]

Step and Delta Functions: \begin{eqnarray*} \frac{d}{dx} \theta(x-a)&=&\delta(x-a)\\ \int_{-\infty}^{\infty} f(x)\delta(x-a)\, dx&=&f(a) \end{eqnarray*}

Vector Calculus Theorems: \begin{eqnarray*} \oint \vec{F} \cdot d\vec{A} &=& \int \vec{\nabla} \cdot \vec{F} d\tau\\ \oint \vec{F} \cdot d\vec{\ell} &=& \int (\vec{\nabla} \times \vec{F}) \cdot d\vec{A}\\ \end{eqnarray*}

Total Charge and Current: \begin{eqnarray*} Q &=& \int \rho (\vec{r}') d\tau'\\ I &=& \int \vec{J} (\vec{r}') \cdot d\vec{A'}\\ \end{eqnarray*}

Students compute a vector line integral, then investigate whether this integral is path independent.

For each of the following vector fields, find a potential function if one exists, or argue that none exists.

• \(\boldsymbol{\vec{F}} = (3x^2 + \tan y)\,\boldsymbol{\hat{x}} + (3y^2 + x\sec^2 y) \,\boldsymbol{\hat{y}}\)
• \(\boldsymbol{\vec{G}} = y\,\boldsymbol{\hat{x}} - x\,\boldsymbol{\hat{y}}\)
• \(\boldsymbol{\vec{H}} = (2xy + y^2 \sin z) \,\boldsymbol{\hat{x}} + (x^2 + z + 2xy\sin z) \,\boldsymbol{\hat{y}} + (y + z + xy^2 \cos z) \,\boldsymbol{\hat{z}}\)
• \(\boldsymbol{\vec{K}} = yz \,\boldsymbol{\hat{x}} + xz \,\boldsymbol{\hat{y}}\)

Main ideas

• Finding potential functions.

Students love this activity. Some groups will finish in 10 minutes or less; few will require as much as 30 minutes. ^*

Prerequisites

• Fundamental Theorem for line integrals
• The Murder Mystery Method

Warmup

none

Props

• whiteboards and pens

Wrapup

• Revisit integrating conservative vector fields along various paths, including reversing the orientation and integrating around closed paths.

Details

In the Classroom

• We recommend having the students work in groups of 2 on this activity, and not having them turn anything in.
• Most students will treat the last example as 2-dimensional, giving the answer \(xyz\). Ask these students to check their work by taking the gradient; most will include a \(\boldsymbol{\hat{z}}\) term. Let them think this through. The correct answer of course depends on whether one assumes that \(z\) is constant; we have deliberately left this ambiguous.
• It is good and proper that students want to add together multivariable terms. Keep returning to the gradient, something they know well. It is better to discover the guidelines themselves.

Subsidiary ideas

• 3-d vector fields do not necessarily have a \(\boldsymbol{\hat{z}}\)-component!

Homework

A challenging question to ponder is why a surface fails to exist for nonconservative fields. Using an example such as \(y\,\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}}\), prompt students to plot the field and examine its magnitude at various locations. Suggest piecing together level sets. There is serious geometry lurking that entails smoothness. Wrestling with this is healthy.

Essay questions

Write 3-5 sentences describing the connection between derivatives and integrals in the single-variable case. In other words, what is the one-dimensional version of MMM? Emphasize that much of vector calculus is generalizing concepts from single-variable theory.

Enrichment

• The derivative check for conservative vector fields can be described using the same type of diagrams as used in the Murder Mystery Method; this is just moving down the diagram (via differentiation) from the row containing the components of the vector field, rather than moving up (via integration). We believe this should not be mentioned until after this lab.

  When done in 3-d, this makes a nice introduction to curl --- which however is not needed until one is ready to do Stokes' Theorem. We would therefore recommend delaying this entire discussion, including the 2-d case, until then.

• Work out the Murder Mystery Method using polar basis vectors, by reversing the process of taking the gradient in this basis.
• Revisit the example in the Ampère's Law lab, using the Fundamental Theorem to explain the results. This can be done without reference to a basis, but it is worth computing \(\boldsymbol{\vec\nabla}\phi\) in a polar basis.

Sketch each of the vector fields below.

1. \(\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
2. \(\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}\)
3. \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)

In this small group activity, students determine various aspects of local points on an elliptic hill which is a function of two variables. The gradient is emphasized as a local quantity which points in the direction of greatest change at a point in the scalar field.

Students set up and compute a scalar surface integral.

Students are asked to review:

• Addition of matrices
• Multiplication of a matrix by a scalar
• Matrix multiplication
• Finding the determinant of a matrix

in preparation for an in-class quiz.

• How to represent 3-d scalar fields in several different ways;
• The symmetries of a some simple charge distributions such as a dipole and a quadrupole.

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.