title, topic, keyword
Small group, whiteboard, etc
Required in-class time for activities
Leave blank to search both

Activities

Small Group Activity

60 min.

Linear Transformations
Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.

Lecture

120 min.

Phase transformations
These lecture notes from the ninth week of https://paradigms.oregonstate.edu/courses/ph441 cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

Inhomogeneous, linear ODEs with constant coefficients are among the most straigtforward to solve, although the algebra can get messy. This content should have been covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see: The Method for Inhomogeneous Equations or your differential equations text.

The general solution of the homogeneous differential equation

\[\ddot{x}-\dot{x}-6 x=0\]

is

\[x(t)=A\, e^{3t}+ B\, e^{-2t}\]

where \(A\) and \(B\) are arbitrary constants that would be determined by the initial conditions of the problem.

  1. Find a particular solution of the inhomogeneous differential equation \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

  2. Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).

  3. Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.

  4. Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)

  5. Find the general solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}-25\sin(4 t)\)

    How is this general solution related to the particular solutions you found in the previous parts of this question?

    Can you add these particular solutions together with arbitrary coefficients to get a new particular solution?

  6. Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.

  • Found in: Oscillations and Waves, None course(s)

Inhomogeneous, linear ODEs with constant coefficients are among the most straigtforward to solve, although the algebra can get messy. This content should have been covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see: The Method for Inhomogeneous Equations or your differential equations text.

For the following inhomogeneous linear equation with constant coefficients, find the general solution for \(y(x)\).

\[y''+2y'-y=\sin{x} +\cos{2x}\]

  • Found in: Oscillations and Waves, None course(s)

Quiz

5 min.

Quiz: Linear Algebra
Overview paragraph here.
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.
Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
  1. Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.

  2. A series of charges arranged in this way is called a linear quadrupole. Why?

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

Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:

Constant Coefficients, Homogeneous

or your differential equations text.

Answer the following questions for each differential equation below:

  • identify the order of the equation,
  • find the number of linearly independent solutions,
  • find an appropriate set of linearly independent solutions, and
  • find the general solution.
Each equation has different notations so that you can become familiar with some common notations.
  1. \(\ddot{x}-\dot{x}-6x=0\)
  2. \(y^{\prime\prime\prime}-3y^{\prime\prime}+3y^{\prime}-y=0\)
  3. \(\frac{d^2w}{dz^2}-4\frac{dw}{dz}+5w=0\)

  • ODEs math bits
    Found in: Oscillations and Waves, None course(s)

Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

  1. Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r}'\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r}'\vert} \] Electrostatic potentials satisfy the superposition principle.

  2. Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

  3. A series of charges arranged in this way is called a linear quadrupole. Why?

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

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.

Small Group Activity

120 min.

Projectile with Linear Drag
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.

Kinesthetic

10 min.

Acting Out Charge Densities
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 we demonstrate the answers with coins under a doc cam.
Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.

Small Group Activity

30 min.

Operators & Functions
Students are asked to:
  • 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.
  • Found in: Quantum Fundamentals course(s)

Kinesthetic

10 min.

Acting Out Current Density
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 steady current. Students act out linear \(\vec{I}\), surface \(\vec{K}\), and volume \(\vec{J}\) current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.

Small Group Activity

30 min.

Expectation Values for a Particle on a Ring
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.

Small Group Activity

10 min.

Modeling Nonuniform Density
In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.
  • Found in: AIMS Maxwell, Static Fields, Surfaces/Bridge Workshop, Problem-Solving course(s)
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.
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.