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

1. Consider the rectangle in the first quadrant of the \(xy\)-plane as in the figure with thick black lines.
   • Label the bottom horizontal edge of the rectangle \(y=c\).
   • Label the sides of the rectangle \(\Delta x\) and \(\Delta y\).
   • What is the area of the rectangle?
   • There are also 2 rectangles whose base is the \(x\)-axis, the larger of which contains both the smaller and the original rectangle. Express the area of the original rectangle as the difference between the areas of these 2 rectangles.
2. On the grid below, draw any simple, closed, piecewise smooth curve \(C\), all of whose segments \(C_i\) are parallel either to the \(x\)-axis or to the \(y\)-axis. Your curve should not be a rectangle. Pick an origin and label it, and assume that each square is a unit square.
   
   • Compute the area of the region \(D\) inside \(C\) by counting the number of squares inside \(C\).

   • Evaluate the line integral \(\displaystyle \oint_C y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) by noticing that along each segment either \(x\) or \(y\) is constant, so that the integral is equal to \(\sum_{C_i} y\,\Delta x\).

     Can you relate this to Problem 1?
     

   • Are your answers to the preceding two calculations the same?
   • Would any of your answers change if you replaced \(y\,\boldsymbol{\hat{x}}\) by \(x\,\boldsymbol{\hat{y}}\) in part (b)?

Main ideas

• Understanding different ways of expressing area using integration.
• Concrete example of Area Corollary to Green's/Stokes' Theorem.

We originally used this activity after covering Green's Theorem; we now skip Green's Theorem and do this activity shortly before Stokes' Theorem.

Prerequisites

• Familiarity with line integrals.
• Green's Theorem is not a prerequisite!

Warmup

• The first problem is a good warmup.

Props

• whiteboards and pens
• a planimeter if available

Wrapup

• Emphasize the magic -- finding area by walking around the boundary!
• Point out that this works for any closed curve, not just the rectangular regions considered here.
• Demonstrate or describe a planimeter, used for instance to measure the area of a region on a map by tracing the boundary.

Details

In the Classroom

• Make sure students use a consistent orientation on their path.
• Make sure students explicitly include all segments of their path, including those which obviously yield zero.
• Students in a given group should all use the same curve.
• Students should be discouraged from drawing a curve whose longest side is along a coordinate axis.
• Students may need to be reminded that \(\oint\) implies the counterclockwise orientation. But it doesn't matter what orientation students use so long as they are consistent!
• A geometric argument that the orientation should be reversed when interchanging \(x\) and \(y\) is to rotate the \(xy\)-plane about the line \(y=x\). (This explains the minus sign in Green's Theorem.)
• Students may not have seen line integrals of this form (see below).
• Students do very well on this lab, particularly after working in groups for several weeks. Resist the urge to intervene.
• Make sure everyone sees the reason \(y\,\boldsymbol{\hat{x}}\cdot d\boldsymbol{\vec{r}}\) is zero on vertical pieces.
• The issue of the negative will come up. Suggest students make a quick sketch of the vector field.
• It is well worthwhile to do an example with a circle together as a class. The line integral should pose no trouble for them and the area of a circle is something they believe.
• Emphasize the connection between the boundary and the interior. This is a concrete display of this relationship.

Subsidiary ideas

• Orientation of closed paths.

• Line integrals of the form \(\int P\,dx+Q\,dy\).

  We do not discuss such integrals in class! Integrals of this form almost always arise in applications as \(\int\boldsymbol{\vec{F}}\cdot d\boldsymbol{\vec{r}}\).

Homework

Determine the area of a triangle or an ellipse by integrating along the boundary.

Essay questions

Describe times in your life when you needed to know area (or imagine such a time). Maybe buying carpet or painting a room. What is the first step in computing area? How does this lab truly differ, if at all?

Enrichment

• Write down Green's Theorem.
• Go to 3 dimensions --- bend the curve out of the plane and stretch the region like a butterfly net or rubber sheet. This is the setting for Stokes' Theorem!

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

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.

• 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.

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.

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.

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.

(4pts) Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation*} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation*}

1. (4pts) Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
2. (4pts) Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.

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!)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

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

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.

• 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 known algebraic expressions for vector line elements \(d\boldsymbol{\vec{r}}\) to determine all simple vector area \(d\boldsymbol{\vec{A}}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

Students compute vector line integrals and explore their properties.

Students compute surface integrals and explore their interpretation as flux.

This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.