assignment_ind Small White Board Question

10 min.

Time Dilation
Theoretical Mechanics (4 years)

Time Dilation Proper Time Special Relativity

Students answer conceptual questions about time dilation and proper time.

accessibility_new Kinesthetic

5 min.

Time Dilation Light Clock Skit

Special Relativity Time Dilation Light Clock Kinesthetic Activity

Students act out the classic light clock scenario for deriving time dilation.

group Small Group Activity

10 min.

Velocity and Acceleration in Polar Coordinates
Central Forces 2023 (3 years) Use geometry to find formulas for velocity and acceleration in polar coordinates.

face Lecture

120 min.

Ideal Gas
Thermal and Statistical Physics 2020

ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.

assignment Homework

Paramagnet (multiple solutions)
Energy and Entropy 2021 (2 years) 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?

group Small Group Activity

30 min.

Paramagnet (multiple solutions)
  • Students evaluate two given partial derivatives from a system of equations.
  • Students learn/review generalized Leibniz notation.
  • Students may find it helpful to use a chain rule diagram.