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.)

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 \(k\).
   • Spherical Symmetry - A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density:

     1. \(\rho (\vec{r}) = \alpha\, r^{3}\)

     2. \(\rho (\vec{r}) =\alpha\, e^{(kr)^{3}}\)

     3. \(\rho (\vec{r}) = \alpha\, \frac{1}{r^{2}}\, e^{(kr)}\)

   • Cylindrical Symmetry - A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density:

     1. \(\rho (\vec{r}) =\alpha\, e^{(ks)^{2}}\)

     2. \(\rho (\vec{r}) = \alpha\, \frac{1}{s}\, e^{(ks)}\)

     3. \(\rho (\vec{r}) = \alpha\, s^{3}\)

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.

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

Students use \(d\boldsymbol{\vec{A} }= d\boldsymbol{\vec{r}}_1 \times d\boldsymbol{\vec{r}}_2\) and \(d\tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\) to find differential surface and volume elements for cylinders and spheres.

Students use completeness relations to write a matrix element of a spin component in a different basis.

For this problem, use the vectors \(|a\rangle = 4 |1\rangle - 3 |2\rangle\) and \(|b\rangle = -i |1\rangle + |2\rangle\).

1. Find \(\langle a | b \rangle\) and \(\langle b | a \rangle\). Discuss how these two inner products are related to each other.
2. For \(\hat{Q}\doteq \begin{pmatrix} 2 & i \\ -i & -2 \end{pmatrix} \), calculate \(\langle1|\hat{Q}|2\rangle\), \(\langle2|\hat{Q}|1\rangle\), \(\langle a|\hat{Q}| b \rangle\) and \(\langle b|\hat{Q}|a \rangle\).
3. What kind of mathematical object is \(|a\rangle\langle b|\)? What is the result if you multiply a ket (for example, \(| a\rangle\) or \(|1\rangle\)) by this expression? What if you multiply this expression by a bra?

1. Let \[|\alpha\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ 1 \end{pmatrix} \qquad \rm{and} \qquad |\beta\rangle \doteq \frac{1}{\sqrt{2}} \begin{pmatrix} 1\\ -1 \end{pmatrix}\] Show that \(\left|{\alpha}\right\rangle \) and \(\left|{\beta}\right\rangle \) are orthonormal. (If a pair of vectors is orthonormal, that suggests that they might make a good basis.)
2. Consider the matrix \[C\doteq \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix} \] Show that the vectors \(|\alpha\rangle\) and \(|\beta\rangle\) are eigenvectors of C and find the eigenvalues. (Note that showing something is an eigenvector of an operator is far easier than finding the eigenvectors if you don't know them!)
3. A operator is always represented by a diagonal matrix if it is written in terms of the basis of its own eigenvectors. What does this mean? Find the matrix elements for a new matrix \(E\) that corresponds to \(C\) expanded in the basis of its eigenvectors, i.e. calculate \(\langle\alpha|C|\alpha\rangle\), \(\langle\alpha|C|\beta\rangle\), \(\langle\beta|C|\alpha\rangle\) and \(\langle\beta|C|\beta\rangle\) and arrange them into a sensible matrix \(E\). Explain why you arranged the matrix elements in the order that you did.
4. Find the determinants of \(C\) and \(E\). How do these determinants compare to the eigenvalues of these matrices?

Students practice infinitesimal reasoning in cylindrical and spherical coordinates.

• How to find area, and volume elements in curvilinear coordinates using geometric methods.

In this small group activity, students multiply a general 3x3 matrix with standard basis row/column vectors to pick out individual matrix elements. Students generate the expressions for the matrix elements in bra/ket notation.

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.

Students are placed into small groups and asked to calculate the total differential of a function of two variables, each of which is in turn expressed in terms of two other variables.

In this unit, you will explore the most common partial differential equations that arise in physics contexts. You will learn the separation of variables procedure to solve these equations.

Motivating Questions

• How are partial differential equations (PDEs) different from ordinary differential equations (ODEs)?
• What new kinds of physics can we learn from solving partial differential equations?
• What can we learn about physics and geometry from the separation of variables procedure?

Key Activities/Problems

• Problem: Laplace's Equation in Polar Coordinates
• Activity: https://paradigms.oregonstate.edu/activity/2084

Unit Learning Outcomes

At the end of this unit, you should be able to:

• Identify and classify several common PDEs.
• Identify the number and types of boundary and initial conditions that are appropriate to different PDEs.
• Identify the conditions where the separation of variables is appropriate and useful.
• Solve simple partial differential equations through the separation of variables procedure.

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.

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Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.