*accessibility_new* Kinesthetic

10 min.

density charge density mass density linear density uniform idealization

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 the we demonstrate the answers with coins under a doc cam.*accessibility_new* Kinesthetic

10 min.

Steady current current density magnetic field idealization

Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of*group* Small Group Activity

5 min.

*accessibility_new* Kinesthetic

10 min.

*groups* Whole Class Activity

10 min.

*computer* Computer Simulation

30 min.

Taylor series power series approximation

Students use a prepared*group* Small Group Activity

120 min.

*group* Small Group Activity

30 min.

Taylor series power series approximation

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

\[c_n={1\over n!}\, f^{(n)}(z_0)\]

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a \(\sin\theta\) (around both the origin and \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*accessibility_new* Kinesthetic

10 min.

*assignment_ind* Small White Board Question

10 min.

Cylindrical coordinates spherical coordinates curvilinear coordinates

First, students are shown diagrams of cylindrical and spherical coordinates. Common notation systems are discussed, especially that physicists and mathematicians use opposite conventions for the angles \(\theta\) and \(\phi\). Then students are asked to check their understanding by sketching several coordinate equals constant surfaces on their small whiteboards.*assignment_ind* Small White Board Question

10 min.

*computer* Mathematica Activity

30 min.

*group* Small Group Activity

30 min.

coulomb's law electric field charge ring symmetry integral power series superposition

Students work in groups of three 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.

*group* Small Group Activity

30 min.

- 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 a Parallel Plate Capacitor” 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: \(\vec{F}=q\vec{E}\)

*group* Small Group Activity

30 min.

*group* Small Group Activity

60 min.

electrostatic potential multipole charge symmetry scalar field superposition coulomb's Law

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.*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula

Students work in groups of three 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.

*group* Small Group Activity

30 min.

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum energy hermitian operators probability superposition representations notations degeneracy

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.*face* Lecture

30 min.

latent heat heat capacity internal energy entropy

This short lecture introduces the ideas required for Ice Calorimetry Lab or Microwave oven Ice Calorimetry Lab.*group* Small Group Activity

120 min.

*group* Small Group Activity

5 min.

Special Relativity Spacetime Diagrams Simultaneity Colocation

Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.*group* Small Group Activity

30 min.

central forces quantum mechanics eigenstates eigenvalues hermitian operators quantum measurements degeneracy expectation values time dependence

Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*grading* Quiz

60 min.

*group* Small Group Activity

30 min.

Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.*group* Small Group Activity

60 min.

Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.*group* Small Group Activity

5 min.

*group* Small Group Activity

60 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

5 min.

*face* Lecture

30 min.

*group* Small Group Activity

30 min.

magnetic fields current Biot-Savart law vector field symmetry

Students work in groups of three 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.

*group* Small Group Activity

30 min.

compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

Students work in groups of three 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.

*group* Small Group Activity

30 min.

energy conservation mass conservation collision

Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

*biotech* Experiment

60 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

E&M Conservative Fields Surfaces

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*group* Small Group Activity

30 min.

- Test to see if one of the given functions is an eigenfunction of the given operator
- See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.

*assignment_ind* Small White Board Question

10 min.

*groups* Whole Class Activity

10 min.

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distribued to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

*assignment_ind* Small White Board Question

10 min.

Special Relativity Spacetime Diagrams Worldlines Postulates of Relativity

Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

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.

*accessibility_new* Kinesthetic

10 min.

Quantum State Vectors Complex Numbers Spin 1/2 Arms Representation

Students, working in pairs, use their left arms to represent each component in a two-state quantum spin 1/2 system. Reinforces the idea that quantum states are complex valued vectors. Students make connections between Dirac, matrix, and Arms representation.*group* Small Group Activity

30 min.

central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.*group* Small Group Activity

10 min.

*accessibility_new* Kinesthetic

30 min.

distance formula coordinate systems dot product vector addition

A short improvisational role-playing skit based on the*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.

central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.*assignment_ind* Small White Board Question

10 min.

*accessibility_new* Kinesthetic

5 min.

Special Relativity Time Dilation Light Clock Kinesthetic Activity

Students act out the classic light clock scenario for deriving time dilation.*group* Small Group Activity

30 min.

*face* Lecture

30 min.

*group* Small Group Activity

30 min.

charge volume charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.*group* Small Group Activity

30 min.

*accessibility_new* Kinesthetic

10 min.

quantum states complex numbers arms Bloch sphere relative phase overall phase Bloch sphere

Students, working in pairs, use the Arms representations to represent states of spin 1/2 system. Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT distinguish quantum states) and relative phase (which does distinguish quantum states).*accessibility_new* Kinesthetic

10 min.

*accessibility_new* Kinesthetic

10 min.

arms complex numbers Argand diagram complex plane rectangular form exponential form complex conjugate

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.*accessibility_new* Kinesthetic

30 min.

*group* Small Group Activity

30 min.

*group* Small Group Activity

10 min.

Arms Sequence for Complex Numbers and Quantum States

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.*assignment_ind* Small White Board Question

10 min.

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..

*group* Small Group Activity

30 min.

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

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

*group* Small Group Activity

30 min.

*computer* Mathematica Activity

30 min.

central forces quantum mechanics angular momentum probability density eigenstates time evolution superposition mathematica

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.*group* Small Group Activity

30 min.

*group* Small Group Activity

30 min.