Circle Trig

  • trigonometry cosine sine math circle
    • group Equipotential Surfaces

      group Small Group Activity

      120 min.

      Equipotential Surfaces

      E&M Quadrupole Scalar Fields

      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.
    • assignment Paramagnet (multiple solutions)

      assignment Homework

      Paramagnet (multiple solutions)
      Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 We have the following equations of state for the total magnetization \(M\), and the entropy \(S\) of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
      1. List variables in their proper positions in the middle columns of the charts below.

      2. Solve for the magnetic susceptibility, which is defined as: \[\chi_B=\left(\frac{\partial M}{\partial B}\right)_T \]

      3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

        \[\left(\frac{\partial M}{\partial B}\right)_S \]

      4. Evaluate your chain rule. Sense-making: Why does this come out to zero?

    • assignment Visualization of Wave Functions on a Ring

      assignment Homework

      Visualization of Wave Functions on a Ring
      Central Forces Spring 2021 Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)
      1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
      2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
      3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
      4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.
    • groups Pineapples and Pumpkins

      groups Whole Class Activity

      10 min.

      Pineapples and Pumpkins
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      Integration Sequence

      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 Flux through a Paraboloid

      assignment Homework

      Flux through a Paraboloid
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.

    • group Electric Field of Two Charged Plates

      group Small Group Activity

      30 min.

      Electric Field of Two Charged Plates
      • 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
        1. objects with like charge repel and opposite charge attract,
        2. object tend to move toward lower energy configurations
        3. The potential energy of a charged particle is related to its charge: \(U=qV\)
        4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
    • group Events on Spacetime Diagrams

      group Small Group Activity

      5 min.

      Events on Spacetime Diagrams
      Theoretical Mechanics Fall 2021

      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.
    • assignment Events on Spacetime Diagrams

      assignment Homework

      Events on Spacetime Diagrams
      Special Relativity Spacetime Diagram Simultaneity Colocation Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021
        1. Which pairs of events (if any) are simultaneous in the unprimed frame?

        2. Which pairs of events (if any) are simultaneous in the primed frame?

        3. Which pairs of events (if any) are colocated in the unprimed frame?

        4. Which pairs of events (if any) are colocated in the primed frame?

      1. For each of the figures, answer the following questions:
        1. Which event occurs first in the unprimed frame?

        2. Which event occurs first in the primed frame?

    • group Vector Surface and Volume Elements

      group Small Group Activity

      30 min.

      Vector Surface and Volume Elements
      AIMS Maxwell AIMS 21

      Integration Sequence

      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 Name the experiment

      group Small Group Activity

      30 min.

      Name the experiment
      Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 Energy and Entropy Fall 2021 Students will design an experiment that measures a specific partial derivative.
  • Quantum Fundamentals Winter 2021

    On the following diagrams, mark both \(\theta\) and \(\sin\theta\) for \(\theta_1=\frac{5\pi}{6}\) and \(\theta_2=\frac{7\pi}{6}\). Write one to three sentences about how these two representations are related to each other. (For example, see: https://phet.colorado.edu/sims/html/trig-tour/latest/trig-tour_en.html)

  • Media & Figures
    • figures/sinewave.png
    • figures/unitcircle.png
    • figures/circletrigsol2.png
    • figures/circletrigsol1.png