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Activities

Small Group Activity

120 min.

##### Equipotential Surfaces
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.

Small Group Activity

30 min.

##### Scalar Surface and Volume Elements

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.

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Integration Sequence sequence(s)

Small Group Activity

30 min.

##### Total Charge
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Integration Sequence sequence(s)

Small Group Activity

60 min.

##### Electrostatic Potential Due to a Pair of Charges (with Series)
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.
• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Power Series Sequence (E&M), E&M Ring Cycle Sequence sequence(s)

Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge

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.

• Found in: Static Fields, AIMS Maxwell, Problem-Solving, Theoretical Mechanics course(s) Found in: Power Series Sequence (E&M), Visualizing Scalar Fields, Warm-Up, E&M Ring Cycle Sequence sequence(s)

Small Group Activity

30 min.

##### The Pretzel
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

• $ds=|d\boldsymbol{\vec{r}}|$
• Found in: Vector Calculus II course(s)

Problem

5 min.

##### Vector Sketch (Rectangular Coordinates)
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. $\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}$
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)

Small Group Activity

30 min.

##### Murder Mystery Method
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

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.
• Found in: Vector Calculus II course(s)

Small Group Activity

60 min.

##### The Wire
Students compute a vector line integral, then investigate whether this integral is path independent.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Workshop Presentations 2023 sequence(s)

Small Group Activity

5 min.

##### Static Fields Equation Sheet

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*}

• Found in: Static Fields course(s)

Small Group Activity

5 min.

##### Acting Out Flux
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.
• Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Gauss/Ampere Sequence (Integral Form), Flux Sequence sequence(s)

Kinesthetic

10 min.

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".
• Found in: Static Fields, AIMS Maxwell, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Gradient Sequence sequence(s)

Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge

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.

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Power Series Sequence (E&M), E&M Ring Cycle Sequence sequence(s)

Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring

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.

• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Power Series Sequence (E&M), E&M Ring Cycle Sequence sequence(s)

Small Group Activity

30 min.

##### Number of Paths
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.

Problem

5 min.

##### Linear Algebra Review/Quiz Prep
• Multiplication of a matrix by a scalar
• Matrix multiplication
• Finding the determinant of a matrix
in preparation for an in-class quiz.

Small Group Activity

30 min.

##### The Cone
Students set up and compute a scalar surface integral.
• Found in: Vector Calculus II, Surfaces/Bridge Workshop course(s) Found in: Workshop Presentations 2023 sequence(s)

Small Group Activity

30 min.

##### Visualization of Curl
Students predict from graphs of simple 2-d vector fields whether the curl is positive, negative, or zero in various regions of the domain using the definition of the curl of a vector field at a point as the maximum circulation per unit area through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence sequence(s)

Small Group Activity

30 min.

##### Visualization of Divergence
Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the geometric definition of the divergence of a vector field at a point as flux per unit volume (here: area) through an infinitesimal box surrounding that point. Optionally, students can use computer algebra to verify their predictions.
• Found in: Static Fields, AIMS Maxwell, Vector Calculus II, Surfaces/Bridge Workshop, Problem-Solving course(s) Found in: Geometry of Vector Fields Sequence, Flux Sequence sequence(s)

Small Group Activity

30 min.

##### The Hill
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.
• Found in: Vector Calculus II, Vector Calculus I, Surfaces/Bridge Workshop course(s) Found in: Gradient Sequence sequence(s)

Mathematica Activity

30 min.

##### Using Technology to Visualize Potentials
Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.
• Found in: Static Fields, AIMS Maxwell, Problem-Solving course(s) Found in: Visualizing Scalar Fields sequence(s)

Small Group Activity

30 min.

##### DELETE Navigating a Hill
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.
• Found in: Static Fields, AIMS Maxwell course(s)

Mathematica Activity

30 min.