assignment Homework

Line Sources Using the Gradient

Gradient Sequence

Static Fields 2022 (5 years)
  1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

keyboard Computational Activity

120 min.

Electrostatic potential of a square of charge
Computational Physics Lab II 2022

integration electrostatic potential surface charge density

Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.

group Small Group Activity

30 min.

Gravitational Force

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.

assignment Homework

Hockey
Central Forces 2023 (3 years)

Consider the frictionless motion of a hockey puck of mass \(m\) on a perfectly circular bowl-shaped ice rink with radius \(a\). The central region of the bowl (\(r < 0.8a\)) is perfectly flat and the sides of the ice bowl smoothly rise to a height \(h\) at \(r = a\).

  1. Draw a sketch of the potential energy for this system. Set the zero of potential energy at the top of the sides of the bowl.
  2. Situation 1: the puck is initially moving radially outward from the exact center of the rink. What minimum velocity does the puck need to escape the rink?
  3. Situation 2: a stationary puck, at a distance \(\frac{a}{2}\) from the center of the rink, is hit in such a way that it's initial velocity \(\vec v_0\) is perpendicular to its position vector as measured from the center of the rink. What is the total energy of the puck immediately after it is struck?
  4. In situation 2, what is the angular momentum of the puck immediately after it is struck?
  5. Draw a sketch of the effective potential for situation 2.
  6. In situation 2, for what minimum value of \(\vec v_0\) does the puck just escape the rink?

group Small Group Activity

30 min.

A glass of water
Energy and Entropy 2021 (2 years)

thermodynamics intensive extensive temperature volume energy entropy

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.

group Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (7 years)

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

Power Series Sequence (E&M)

Ring Cycle Sequence

Warm-Up

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.

assignment Homework

Scattering
Central Forces 2023 (3 years)

Consider a very light particle of mass \(\mu\) scattering from a very heavy, stationary particle of mass \(M\). The force between the two particles is a repulsive Coulomb force \(\frac{k}{r^2}\). The impact parameter \(b\) in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass \(\mu\) is \(\vec v_0\). Answer the following questions about this situation in terms of \(k\), \(M\), \(\mu\), \(\vec v_0\), and \(b\). ()It is not necessarily wise to answer these questions in order.)

  1. What is the initial angular momentum of the system?
  2. What is the initial total energy of the system?
  3. What is the distance of closest approach \(r_{\rm{min}}\) with the interaction?
  4. Sketch the effective potential.
  5. What is the angular momentum at \(r_{\rm{min}}\)?
  6. What is the total energy of the system at \(r_{\rm{min}}\)?
  7. What is the radial component of the velocity at \(r_{\rm{min}}\)?
  8. What is the tangential component of the velocity at \(r_{\rm{min}}\)?
  9. What is the value of the effective potential at \(r_{\rm{min}}\)?
  10. For what values of the initial total energy are there bound orbits?
  11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

assignment Homework

Line Sources Using Coulomb's Law
Static Fields 2022 (5 years)
  1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
  2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.

face Lecture

120 min.

Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

chemical potential Gibbs distribution grand canonical ensemble statistical mechanics

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.

keyboard Computational Activity

120 min.

Kinetic energy
Computational Physics Lab II 2022

finite difference hamiltonian quantum mechanics particle in a box eigenfunctions

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.

group Small Group Activity

60 min.

Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (5 years)

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

Power Series Sequence (E&M)

Ring Cycle Sequence

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.

face Lecture

120 min.

Phase transformations
Thermal and Statistical Physics 2020

phase transformation Clausius-Clapeyron mean field theory thermodynamics

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.

assignment Homework

Ideal gas in two dimensions
Ideal gas Entropy Chemical potential Thermal and Statistical Physics 2020
  1. Find the chemical potential of an ideal monatomic gas in two dimensions, with \(N\) atoms confined to a square of area \(A=L^2\). The spin is zero.

  2. Find an expression for the energy \(U\) of the gas.

  3. Find an expression for the entropy \(\sigma\). The temperature is \(kT\).