assignment Homework

Gravitational Field and Mass
Static Fields 2022 (5 years)

The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}

This problem explores the consequences of the divergence theorem for this shell.

  1. Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
  2. Briefly discuss the physical meaning of the divergence in this particular example.
  3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\). ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
  4. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

assignment Homework

Magnetic susceptibility
Paramagnet Magnetic susceptibility Thermal and Statistical Physics 2020

Consider a paramagnet, which is a material with \(n\) spins per unit volume each of which may each be either “up” or “down”. The spins have energy \(\pm mB\) where \(m\) is the magnetic dipole moment of a single spin, and there is no interaction between spins. The magnetization \(M\) is defined as the total magnetic moment divided by the total volume. Hint: each individual spin may be treated as a two-state system, which you have already worked with above.

Plot of magnetization vs. B field

  1. Find the Helmholtz free energy of a paramagnetic system (assume \(N\) total spins) and show that \(\frac{F}{NkT}\) is a function of only the ratio \(x\equiv \frac{mB}{kT}\).

  2. Use the canonical ensemble (i.e. partition function and probabilities) to find an exact expression for the total magentization \(M\) (which is the total dipole moment per unit volume) and the susceptibility \begin{align} \chi\equiv\left(\frac{\partial M}{\partial B}\right)_T \end{align} as a function of temperature and magnetic field for the model system of magnetic moments in a magnetic field. The result for the magnetization is \begin{align} M=nm\tanh\left(\frac{mB}{kT}\right) \end{align} where \(n\) is the number of spins per unit volume. The figure shows what this magnetization looks like.

  3. Show that the susceptibility is \(\chi=\frac{nm^2}{kT}\) in the limit \(mB\ll kT\).

computer Mathematica Activity

30 min.

Using Technology to Visualize Potentials
Static Fields 2022 (6 years)

electrostatic potential visualization

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made Sage code or a Mathematica worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrates several different ways of plotting the potential.

assignment Homework

Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2022 (4 years) Consider the electric field \begin{equation} \vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases} \end{equation}
  1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
  2. Find a formula for the charge density that creates this electric field.
  3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.

group Small Group Activity

30 min.

Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (6 years)

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

Power Series Sequence (E&M)

Ring Cycle Sequence

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.

face Lecture

5 min.

Central Forces Introduction: Lecture Notes
Central Forces 2023 (2 years)

assignment Homework

Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

  1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
  2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

group Small Group Activity

60 min.

Gravitational Potential Energy

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.

keyboard Computational Activity

120 min.

Electrostatic potential of spherical shell
Computational Physics Lab II 2022

electrostatic potential spherical coordinates

Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize.

computer Mathematica Activity

30 min.

Visualization of Quantum Probabilities for the Hydrogen Atom
Central Forces 2023 (3 years) Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).

accessibility_new Kinesthetic

10 min.

Acting Out Charge Densities
Static Fields 2022 (6 years)

density charge density mass density linear density uniform idealization

Ring Cycle Sequence

Integration Sequence

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.

group Small Group Activity

30 min.

Total Charge
Static Fields 2022 (6 years)

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

Integration Sequence

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

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.

accessibility_new Kinesthetic

10 min.

Curvilinear Basis Vectors
Static Fields 2022 (9 years)

symmetry curvilinear coordinate systems basis vectors

Curvilinear Coordinate Sequence

Students use their arms to depict (sequentially) the different cylindrical and spherical basis vectors at the location of their shoulder (seen in relation to a specified origin of coordinates: either a set of axes hung from the ceiling of the room or perhaps a piece of furniture or a particular corner of the room).