assignment Homework

Dimensional Analysis of Kets
dirac notation dimensions probability completeness relations

Completeness Relations

  1. \(\left\langle {\Psi}\middle|{\Psi}\right\rangle =1\) Identify and discuss the dimensions of \(\left|{\Psi}\right\rangle \).
  2. For a spin \(\frac{1}{2}\) system, \(\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1\). Identify and discuss the dimensions of \(\left|{+}\right\rangle \) and \(\left|{-}\right\rangle \).
  3. In the position basis \(\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1\). Identify and discuss the dimesions of \(\left|{x}\right\rangle \).

group Small Group Activity

30 min.

Black space capsule
Contemporary Challenges 2022 (3 years)

stefan-boltzmann blackbody radiation

In this activity, students apply the Stefan-Boltzmann equation and the principle of energy balance in steady state to find the steady state temperature of a black object in near-Earth orbit.

assignment Homework

Total Current, Circular Cross Section

Integration Sequence

Static Fields 2023 (5 years)

A current \(I\) flows down a cylindrical wire of radius \(R\).

  1. If it is uniformly distributed over the surface, give a formula for the surface current density \(\vec K\).
  2. If it is distributed in such a way that the volume current density, \(|\vec J|\), is inversely proportional to the distance from the axis, give a formula for \(\vec J\).

assignment Homework

Linear Quadrupole (w/o series)
Static Fields 2023 (4 years) 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?

accessibility_new Kinesthetic

10 min.

Acting Out the Gradient
Static Fields 2023 (6 years)

gradient vector fields electrostatics

Gradient Sequence

Students are shown a topographic map of an oval hill and imagine that the classroom is on the hill. They are asked to point in the direction of the gradient vector appropriate to the point on the hill where they are "standing".

assignment Homework

Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2023 (6 years)

For each case below, find the total charge.

  1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
  2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}

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!)

assignment Homework

Mass of a Slab
Static Fields 2023 (6 years)

Determine the total mass of each of the slabs below.

  1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
  2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
  3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
  4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
  5. What are the dimensions of \(A\)?
  6. Write several sentences comparing your answers to the different cases above.

computer Mathematica Activity

30 min.

Visualising the Gradient
Static Fields 2023 (6 years)

Gradient Sequence

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

assignment Homework

Cube Charge
charge density

Integration Sequence

Static Fields 2023 (6 years)
  1. Charge is distributed throughout the volume of a dielectric cube with charge density \(\rho=\beta z^2\), where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
  2. On a different cube: Charge is distributed on the surface of a cube with charge density \(\sigma=\alpha z\) where \(z\) is the height from the bottom of the cube, and where each side of the cube has length \(L\). What is the total charge on the cube? Don't forget about the top and bottom of the cube.

assignment Homework

Using Gibbs Free Energy
thermodynamics entropy heat capacity internal energy equation of state Energy and Entropy 2021 (2 years)

You are given the following Gibbs free energy: \begin{equation*} G=-k T N \ln \left(\frac{a T^{5 / 2}}{p}\right) \end{equation*} where \(a\) is a constant (whose dimensions make the argument of the logarithm dimensionless).

  1. Compute the entropy.

  2. Work out the heat capacity at constant pressure \(C_p\).

  3. Find the connection among \(V\), \(p\), \(N\), and \(T\), which is called the equation of state (Hint: find the volume as a partial derivative of the Gibbs free energy).

  4. Compute the internal energy \(U\).

accessibility_new Kinesthetic

10 min.

Acting Out Charge Densities
Static Fields 2023 (6 years)

density charge density mass density linear density uniform idealization

Integration Sequence

Ring Cycle 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 2023 (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.

group Small Group Activity

60 min.

Going from Spin States to Wavefunctions
Quantum Fundamentals 2023 (2 years)

Wavefunctions quantum states probability amplitude histograms matrix notation of quantum states Arms representation

Arms Sequence for Complex Numbers and Quantum States

Completeness Relations

Students review using the Arms representation to represent states for discrete quantum systems and connecting the Arms representation to histogram and matrix representation. The student then extend the Arms representation to begin exploring the continuous position basis.

assignment Homework

Differentials of One Variable
Static Fields 2023 (6 years) Find the total differential of the following functions:
  1. \(y=3x^2 + 4\cos 2x\)
  2. \(y=3x^2\cos kx\) (where \(k\) is a constant)
  3. \(y=\frac{\cos 7x}{x^2}\)
  4. \(y=\cos(3x^2-2)\)

group Small Group Activity

30 min.

de Broglie wavelength after freefall
Contemporary Challenges 2021 (4 years)

de Broglie wavelength gravity

In this activity students combine energy conservation with the relationship between the de Broglie wavelength and momentum to find the wavelength of atoms that have been dropped a given distance.

assignment Homework

Pressure of thermal radiation
Thermal radiation Pressure Thermal and Statistical Physics 2020

(modified from K&K 4.6) We discussed in class that \begin{align} p &= -\left(\frac{\partial F}{\partial V}\right)_T \end{align} Use this relationship to show that

  1. \begin{align} p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right), \end{align} where \(\langle n_j\rangle\) is the number of photons in the mode \(j\);

  2. Solve for the relationship between pressure and internal energy.

keyboard Computational Activity

120 min.

Mean position
Computational Physics Lab II 2023 (2 years)

probability density particle in a box wave function quantum mechanics

Students compute probabilities and averages given a probability density in one dimension. This activity serves as a soft introduction to the particle in a box, introducing all the concepts that are needed.

accessibility_new Kinesthetic

10 min.

Acting Out Current Density
Static Fields 2023 (6 years)

Steady current current density magnetic field idealization

Integration Sequence

Ring Cycle Sequence

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.

group Small Group Activity

120 min.

Projectile with Linear Drag
Theoretical Mechanics (4 years)

Projectile Motion Drag Forces Newton's 2nd Law Separable Differential Equations

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.