Activities
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 we demonstrate the answers with coins under a doc cam.
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.
Students explore what linear transformation matrices do to vectors. The whole class discussion compares & contrasts several different types of transformations (rotation, flip, projections, “scrinch”, scale) and how the properties of the matrices (the determinant, symmetries, which vectors are unchanged) are related to these transformations.
Problem
You have a charge distribution on the \(x\)-axis composed of two point charges: one with charge \(+3q\) located at \(x=-d\) and the other with charge \(-q\) located at \(x=+d\).
- Sketch the charge distribution.
- Write an expression for the volume charge density \(\rho (\vec{r})\) everywhere in space.
Students work in small groups 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.
Problem
Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.
A series of charges arranged in this way is called a linear quadrupole. Why?
Consider a rod of length \(L\) lying on the \(z\)-axis. Find an algebraic expression for the mass density of the rod if the mass density at \(z=0\) is \(\lambda_0\) and at \(z=L\) is \(7\lambda_0\) and you know that the mass density increases
- (2pts) linearly;
- (2pts) like the square of the distance along the rod;
- (2pts) exponentially.
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}
- (4pts) Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
- (4pts) Find a formula for the charge density that creates this electric field.
- (2pts) Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
Consider a white dwarf of mass \(M\) and radius \(R\). The dwarf consists of ionized hydrogen, thus a bunch of free electrons and protons, each of which are fermions. Let the electrons be degenerate but nonrelativistic; the protons are nondegenerate.
Show that the order of magnitude of the gravitational self-energy is \(-\frac{GM^2}{R}\), where \(G\) is the gravitational constant. (If the mass density is constant within the sphere of radius \(R\), the exact potential energy is \(-\frac53\frac{GM^2}{R}\)).
Show that the order of magnitude of the kinetic energy of the electrons in the ground state is \begin{align} \frac{\hbar^2N^{\frac53}}{mR^2} \approx \frac{\hbar^2M^{\frac53}}{mM_H^{\frac53}R^2} \end{align} where \(m\) is the mass of an electron and \(M_H\) is the mas of a proton.
Show that if the gravitational and kinetic energies are of the same order of magnitude (as required by the virial theorem of mechanics), \(M^{\frac13}R \approx 10^{20} \text{g}^{\frac13}\text{cm}\).
If the mass is equal to that of the Sun (\(2\times 10^{33}g\)), what is the density of the white dwarf?
It is believed that pulsars are stars composed of a cold degenerate gas of neutrons (i.e. neutron stars). Show that for a neutron star \(M^{\frac13}R \approx 10^{17}\text{g}^{\frac13}\text{cm}\). What is the value of the radius for a neutron star with a mass equal to that of the Sun? Express the result in \(\text{km}\).
Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state.
Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.
For each case below, find the total charge.
- (4pts) 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*}
- (4pts) 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*}
Problem
Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r}'\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r}'\vert} \] Electrostatic potentials satisfy the superposition principle.
Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.
- A series of charges arranged in this way is called a linear quadrupole. Why?
Homogeneous, linear ODEs with constant coefficients were likely covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see:
Constant Coefficients, Homogeneous
or your differential equations text.
Answer the following questions for each differential equation below:
Each equation has different notations so that you can become familiar with some common notations.
- identify the order of the equation,
- find the number of linearly independent solutions,
- find an appropriate set of linearly independent solutions, and
- find the general solution.
- \(\ddot{x}-\dot{x}-6x=0\)
- \(y^{\prime\prime\prime}-3y^{\prime\prime}+3y^{\prime}-y=0\)
- \(\frac{d^2w}{dz^2}-4\frac{dw}{dz}+5w=0\)
Consider the volume charge density: \begin{equation*} \rho (x,y,z)=c\,\delta (x-3) \end{equation*}
- (2 pts) Describe in words how this charge is distributed in space.
- (2 pts) What are the dimensions of the constant \(c\)?
Students are asked to review:in preparation for an in-class quiz.
- Addition of matrices
- Multiplication of a matrix by a scalar
- Matrix multiplication
- Finding the determinant of a matrix
Inhomogeneous, linear ODEs with constant coefficients are among the most straigtforward to solve, although the algebra can get messy. This content should have been covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see: The Method for Inhomogeneous Equations or your differential equations text.
For the following inhomogeneous linear equation with constant coefficients, find the general solution for \(y(x)\).
\[y''+2y'-y=\sin{x} +\cos{2x}\]
Inhomogeneous, linear ODEs with constant coefficients are among the most straigtforward to solve, although the algebra can get messy. This content should have been covered in your Differential Equations course (MTH 256 or equiv.). If you need a review, please see: The Method for Inhomogeneous Equations or your differential equations text.
The general solution of the homogeneous differential equation
\[\ddot{x}-\dot{x}-6 x=0\]
is
\[x(t)=A\, e^{3t}+ B\, e^{-2t}\]
where \(A\) and \(B\) are arbitrary constants that would be determined by the initial conditions of the problem.
Find a particular solution of the inhomogeneous differential equation \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).
Find the general solution of \(\ddot{x}-\dot{x}-6 x=-25\sin(4 t)\).
Some terms in your general solution have an undetermined coefficients, while some coefficients are fully determined. Explain what is different about these two cases.
Find a particular solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}\)
Find the general solution of \(\ddot{x}-\dot{x}-6 x=12 e^{-3 t}-25\sin(4 t)\)
How is this general solution related to the particular solutions you found in the previous parts of this question?
Can you add these particular solutions together with arbitrary coefficients to get a new particular solution?
- Sense-making: Check your answer; Explicitly plug in your final answer in part (e) and check that it satisfies the differential equation.
Problem
- 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.
- 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.
Mathematica Activity
30 min.
Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.
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.
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.
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
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.
A pretzel is to be dipped in chocolate. The pretzel is in the shape of a quarter circle, consisting of a straight segment from the origin to the point (2,0), a circular arc from there to (0,2), followed by a straight segment back to the origin; all distances are in centimeters. The (linear) density of chocolate on the pretzel is given by \(\lambda = 3(x^ 2 + y^2 )\) in grams per centimeter. Find the total amount of chocolate on the pretzel.Main ideas
- Calculating (scalar) line integrals.
- Use what you know!
Prerequisites
- Familiarity with \(d\boldsymbol{\vec{r}}\).
- Familiarity with “Use what you know” strategy.
Warmup
It is not necessary to explicitly introduce scalar line integrals, before this lab; figuring out that the (scalar) line element must be \(|d\boldsymbol{\vec{r}}|\) can be made part of the activity (if time permits).
Props
- whiteboards and pens
- “linear” chocolate covered candy (e.g. Pocky)
Wrapup
Emphasize that students must express each integrand in terms of a single variable prior to integration.
Emphasize that each integral must be positive!
Discuss several different ways of doing this problem (see below).
Details
In the Classroom
- Make sure the shape of the pretzel is clear! It might be worth drawing it on the board.
- Some students will work geometrically, determining \(ds\) on each piece by inspection. This is fine, but encourage such students to try using \(d\vec{r}\) afterwards.
- Polar coordinates are natural for all three parts of this problem, not just the circular arc.
- Many students will think that the integral “down” the \(y\)-axis should be negative. They will argue that \(ds=dy\), but the limits are from \(2\) to \(0\). The resolution is that \(ds = |dy\,\boldsymbol{\hat x}|=|dy|=-dy\) when integrating in this direction.
- Unlike work or circulation, the amount of chocolate does not depend on which way one integrates, so there is in fact no need to integrate “down” the \(y\)-axis at all.
- Some students may argue that \(d\boldsymbol{\vec{r}}=\boldsymbol{\hat T}\,ds\Longrightarrow ds=d\boldsymbol{\vec{r}}\cdot\boldsymbol{\hat T}\), and use this to get the signs right. This is fine if it comes up, but the unit tangent vector \(\boldsymbol{\hat T}\) is not a fundamental part of our approach.
- There is of course a symmetry argument which says that the two “legs” along the axes must have the same amount of chocolate --- although some students will put a minus sign into this argument!
Subsidiary ideas
- \(ds=|d\boldsymbol{\vec{r}}|\)
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
One way to write volume charge densities without using piecewise functions is to use step \((\Theta)\) or \(\delta\) functions. Consider a spherical shell with charge density \[\rho (\vec{r})=\alpha3e^{(k r)^3} \]
between the inner radius \(a\) and the outer radius \(b\). The charge density is zero everywhere else.
- (2 pts) What are the dimensions of the constants \(\alpha\) and \(k\)?
- (2 pts) By hand, sketch a graph the charge density as a function of \(r\) for \(\alpha > 0\) and \(k>0\) .
- (2 pts) Use step functions to write this charge density as a single function valid everywhere in space.
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
Students calculate the expectation value of energy and angular momentum as a function of time for an initial state for a particle on a ring. This state is a linear combination of energy/angular momentum eigenstates written in bra-ket notation.
Kinesthetic
30 min.
Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.
Students are asked to:
- Test to see if one of the given functions is an eigenfunction of the given operator
- See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
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 \(m\).
Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.
Students find a wavefunction that corresponds to a Gaussian probability density.