These notes from the fifth week of https://paradigms.oregonstate.edu/courses/ph441 cover the grand canonical ensemble. They include several small group activities.

These notes from week 6 of https://paradigms.oregonstate.edu/courses/ph441 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.

The concentration of potassium \(\text{K}^+\) ions in the internal sap of a plant cell (for example, a fresh water alga) may exceed by a factor of \(10^4\) the concentration of \(\text{K}^+\) ions in the pond water in which the cell is growing. The chemical potential of the \(\text{K}^+\) ions is higher in the sap because their concentration \(n\) is higher there. Estimate the difference in chemical potential at \(300\text{K}\) and show that it is equivalent to a voltage of \(0.24\text{V}\) across the cell wall. Take \(\mu\) as for an ideal gas. Because the values of the chemical potential are different, the ions in the cell and in the pond are not in diffusive equilibrium. The plant cell membrane is highly impermeable to the passive leakage of ions through it. Important questions in cell physics include these: How is the high concentration of ions built up within the cell? How is metabolic energy applied to energize the active ion transport?

David adds

You might wonder why it is even remotely plausible to consider the ions in solution as an ideal gas. The key idea here is that the ideal gas entropy incorporates the entropy due to position dependence, and thus due to concentration. Since concentration is what differs between the cell and the pond, the ideal gas entropy describes this pretty effectively. In contrast to the concentration dependence, the temperature-dependence of the ideal gas chemical potential will not be so great.

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 \(kT\).

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.

This is a great first programming activity.

Consider a system consisting of four point charges arranged on the corners of a square in 3D Cartesian space of coordinates \((x,y,z)\).

1. Write a python function that returns the potential at any point in space caused by four equal point charges forming a square. Make the sides of the square parallel to the \(x\) and \(y\) axes and on the \(z=0\) plane.

   To do this you will need the expression for a the potential due to a single point charge \(V= \frac{k_Cq}{r}\) where \(r\) is the distance from the point charge. You will also need to use the fact that the total potential is the sum of the potentials due to each individual point charge.

   It is important that we ask students first to create a function for the potential, and only then try to visualize the potential. This allows students to reason about the computation for a single point in space (defined in their choice of coordinate systems).

2. Once you have written the above function, use it to plot the electrostatic potential versus position along the three cartesian axes.

   Since the students have already written a function for their potential, they can create a plot by creating an array for \(x\) (or \(y\), or \(z\)), and then passing that array to their function, along with scalars for the other two coordinates. Many students will discover this simply by modifying an example script they find on the web, replacing \(\sin(x)\) or similar with their function. It is well worth showing this easier approach to students who attempt who attempt to write a loop in order to compute the potential at each point in space.

   We ask students to explicitly plot the potential along axes because students seldom spontaneously think to create a 1D plot such as this.

3. Label your axes.
4. Work out the first non-zero term in a power series approximation for the potential at large \(x\), small \(x\), etc. Plot these approximations along with your computed potential, and verify that they agree in the range that you expect. Useful 1\(^{st}\) order Taylor expansions are: \begin{eqnarray} \sqrt{1+\epsilon} &\sim& 1+\frac{\epsilon}{2} \\ \frac{1}{1+\epsilon} &=& 1-\epsilon \end{eqnarray} where \(\epsilon\) is a small quantity.

   This may need to be omitted on the first Tuesday of class, since students probably will not yet have seen power series approximations. It may work in this case to at least talk about what is expected at large distance, since "it looks like a point charge" is reasoning students do make.

   Students struggle with the \(x\) approximations (assuming the square is in the xy plane). Each pair will probably need to have a little lecture on grouping terms according to the power of \(x\), and keeping only those terms for which they have every instance.

Extra fun

Create one or more different visualizations of the electrostatic potential. For example a 2D representation in the \(z=0\) plane.

More extra fun

Create a plot of the potential along a straight line that is not one of the axes. Hint: start from a line on the \(z=0\) plane, then try a random straight line. You can use your browser for help.

Even more extra fun

Move the charges around (e.g., off the \(z=0\) plane) and see what happens to your graphs

Dipole fun

Repeat the above (especially the limiting cases!) for four point charges in which half are positive and half negative, with the positive charges neighbors.

Common visualizations for 2D slices of space include contour plots, color plots, and "3D plots". Another option (less easy) would be to visualize an equipotential surface in 3 dimensions. It is worth reminding students to consider other planes than those at \(x=0\), \(y=0\), and \(z=0\).

Quadrupole fun

Repeat the above (especially the limiting cases!) for four point charges in which half are positive and half negative, with the positive charges diagonal from one another. It will help in this case to place the charges on the axes (rotating the square by 45 degrees), since otherwise the potential on each axis will be zero.

None

Consider a column of atoms each of mass \(M\) at temperature \(T\) in a uniform gravitational field \(g\). Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom \(h=0\) of the column. Integrate from \(h=0\) to \(h=\infty\). You may assume the gas is ideal.

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.

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.

Students work in small groups 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.

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). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.

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.

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.

See also the following more detailed problem and solution: Effective Potentials: Graphical Version

An electron is moving on a two dimension surface with a radially symmetric electrostatic potential given by the graph below:

1. Sketch the effective potential if the angular momentum is not zero.
2. Describe qualitatively, the shapes of all possible types of orbits, indicating the energy for each in your diagram.

The electrostatic potential due to a point charge at the origin is given by: \begin{equation*} V=\frac{1}{4\pi\epsilon_0} \frac{q}{r} \end{equation*}

1. (2pts) Find the electric field due to a point charge at the origin as a gradient in rectangular coordinates.
2. (2pts) Find the electric field due to a point charge at the origin as a gradient in spherical coordinates.
3. (2pts) Find the electric field due to a point charge at the origin as a gradient in cylindrical coordinates.

(4pts) 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*}

A circular cylinder of radius \(R\) rotates about the long axis with angular velocity \(\omega\). The cylinder contains an ideal gas of atoms of mass \(M\) at temperature \(T\). Find an expression for the dependence of the concentration \(n(r)\) on the radial distance \(r\) from the axis, in terms of \(n(0)\) on the axis. Take \(\mu\) as for an ideal gas.

For each of the following vector fields, find a potential function if one exists, or argue that none exists.

• \(\boldsymbol{\vec{F}} = (3x^2 + \tan y)\,\boldsymbol{\hat{x}} + (3y^2 + x\sec^2 y) \,\boldsymbol{\hat{y}}\)
• \(\boldsymbol{\vec{G}} = y\,\boldsymbol{\hat{x}} - x\,\boldsymbol{\hat{y}}\)
• \(\boldsymbol{\vec{H}} = (2xy + y^2 \sin z) \,\boldsymbol{\hat{x}} + (x^2 + z + 2xy\sin z) \,\boldsymbol{\hat{y}} + (y + z + xy^2 \cos z) \,\boldsymbol{\hat{z}}\)
• \(\boldsymbol{\vec{K}} = yz \,\boldsymbol{\hat{x}} + xz \,\boldsymbol{\hat{y}}\)

Main ideas

• Finding potential functions.

Students love this activity. Some groups will finish in 10 minutes or less; few will require as much as 30 minutes. ^*

Prerequisites

• Fundamental Theorem for line integrals
• The Murder Mystery Method

Warmup

none

Props

• whiteboards and pens

Wrapup

• Revisit integrating conservative vector fields along various paths, including reversing the orientation and integrating around closed paths.

Details

In the Classroom

• We recommend having the students work in groups of 2 on this activity, and not having them turn anything in.
• Most students will treat the last example as 2-dimensional, giving the answer \(xyz\). Ask these students to check their work by taking the gradient; most will include a \(\boldsymbol{\hat{z}}\) term. Let them think this through. The correct answer of course depends on whether one assumes that \(z\) is constant; we have deliberately left this ambiguous.
• It is good and proper that students want to add together multivariable terms. Keep returning to the gradient, something they know well. It is better to discover the guidelines themselves.

Subsidiary ideas

• 3-d vector fields do not necessarily have a \(\boldsymbol{\hat{z}}\)-component!

Homework

A challenging question to ponder is why a surface fails to exist for nonconservative fields. Using an example such as \(y\,\boldsymbol{\hat{x}}+\boldsymbol{\hat{y}}\), prompt students to plot the field and examine its magnitude at various locations. Suggest piecing together level sets. There is serious geometry lurking that entails smoothness. Wrestling with this is healthy.

Essay questions

Write 3-5 sentences describing the connection between derivatives and integrals in the single-variable case. In other words, what is the one-dimensional version of MMM? Emphasize that much of vector calculus is generalizing concepts from single-variable theory.

Enrichment

• The derivative check for conservative vector fields can be described using the same type of diagrams as used in the Murder Mystery Method; this is just moving down the diagram (via differentiation) from the row containing the components of the vector field, rather than moving up (via integration). We believe this should not be mentioned until after this lab.

  When done in 3-d, this makes a nice introduction to curl --- which however is not needed until one is ready to do Stokes' Theorem. We would therefore recommend delaying this entire discussion, including the 2-d case, until then.

• Work out the Murder Mystery Method using polar basis vectors, by reversing the process of taking the gradient in this basis.
• Revisit the example in the Ampère's Law lab, using the Fundamental Theorem to explain the results. This can be done without reference to a basis, but it is worth computing \(\boldsymbol{\vec\nabla}\phi\) in a polar basis.

Write the equation for the electrostatic potential due to a point charge.

Instructor's Guide

Prerequisite Knowledge

Students will usually have seen the electrostatic potential due to a point charge in their introductory course, but may have trouble recalling it.

Whole-Class Conversations

As students try to remember the formula, many will conflate potential, potential energy, force, and electric field. Their answers may have some aspects of each of these. We use this question to get the iconic equation into the students' working memory in preparation for subsequent activities. This question also be used to help student disambiguate these different physical quantities.

Correct answers you're likely to see

\[V=\frac{kq}{r}\]

\[V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}\] You may want to discuss which constants to use in which contexts, e.g. \(k\) is short and easy to write, but may be conflated with other uses of \(k\) in a give problem whereas \(\frac{1}{4\pi\epsilon_0}\) assumes you are working in a particular system of units.

Incorrect answers you're likely to see

• Two charges instead of one \[\cancel{V=\frac{kq_{1}q_{2}}{r}}\]
• Distance squared in the denominator \[\cancel{V=\frac{kq}{r^2}}\]
• Vector values \[\cancel{V=\frac{kq\, \hat r}{r}}\]

Possible follow-up questions to help with the disambiguation:

• Relationship between potential and potential energy \(U = qV\)
• Which function is the derivative of the other: \(1/r\) or \(1/r^2\)?
• Which physical quantity (potential or electric field, potential energy or force) is the derivative of the other?
• What is the electrostatic potential conceptually?
• Which function falls off faster: \(1/r\) or \(1/r^2\)?
• What are the dimensions of potential? Units?
• Where is the zero of potential?

Wrap-up

• This could be a good time to refer to the (correct) expression for the potential as an iconic equation, which will need to be further interpreted (”unpacked”) in particular physical situations. This is where the course is going next.
• This SWBQ can also serve to help students learn about recall as a cognitive activity. While parts of the equations that students write may be incorrect, many other parts will be correct. Let the way in which you manage the class discussion model for the students how a professional goes about quickly disambiguating several different choices. And TELL the students that this is what you are doing. Deliberately invoke their metacognition.
• Many students may not know that the electrostatic potential that we are talking about in this activity is the same quantity as what a voltmeter reads, in principle, but not in practice. You may need to talk about how a voltmeter actually works, rather than idealizing it. It helps to have a voltmeter with leads as a prop. Students often want to know about the “ground” lead. We often tie a long string to it (to symbolize making a really long wire) and send the TA out of the room with the string, “headed off to infinity” while discussing the importance of setting the zero of potential. The extra minute or two of humerous byplay gives the importance of the zero of potential a chance to sink in.

We use this small whiteboard question as a transition between The Distance Formula (Star Trek) activity, where students are learning about how to describe (algebraically) the geometric distance between two points, and the Electrostatic Potential Due to a Pair of Charges (with Series) activity, where students are using these results and the superposition principle to find the electrostatic potential due to two point charges.

This activity is the initial activity in the sequence Visualizing Scalar Fields addressing the representations of scalar fields in the context of electrostatics.

• How to represent 3-d scalar fields in several different ways;
• The symmetries of a some simple charge distributions such as a dipole and a quadrupole.

A student is invited to “act out” motion corresponding to a plot of effective potential vs. distance. The student plays the role of the “Earth” while the instructor plays the “Sun”.

Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.

Students observe the motion of a puck tethered to the center of the airtable. Then they plot the potential energy for the puck on their small whiteboards. A class discussion follows based on what students have written on their whiteboards.

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.

Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

• Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
• Students should know that
  1. objects with like charge repel and opposite charge attract,
  2. object tend to move toward lower energy configurations
  3. The potential energy of a charged particle is related to its charge: \(U=qV\)
  4. The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

Students use a PhET simulation to explore the time evolution of a particle in an infinite square well potential.