assignment Homework

Gibbs free energy
thermodynamics Maxwell relation Energy and Entropy 2020 The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
  1. Find the total differential of \(G\). As always, show your work.
  2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
  3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

assignment Homework

Approximating a Delta Function with Isoceles Triangles
Static Fields 2022 (4 years)

Remember that the delta function is defined so that \[ \delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases} \]

Also: \[\int_{-\infty}^{\infty} \delta(x-a)\, dx =1\].

  1. Find a set of functions that approximate the delta function \(\delta(x-a)\) with a sequence of isosceles triangles \(\delta_{\epsilon}(x-a)\), centered at \(a\), that get narrower and taller as the parameter \(\epsilon\) approaches zero.
  2. Using the test function \(f(x)=3x^2\), find the value of \[\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx\] Then, show that the integral approaches \(f(a)\) in the limit that \(\epsilon \rightarrow 0\).

group Small Group Activity

5 min.

Acting Out Flux
Static Fields 2022 (3 years)

flux electrostatics vector fields

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

assignment Homework

Contours
Static Fields 2022 (4 years)

Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\) and \(y\) are measured in meters and that \(\mu\) is measured in kilograms. Four points are indicated on the plot.

  1. Determine \(\frac{\partial\mu}{\partial x}\) and \(\frac{\partial\mu}{\partial y}\) at each of the four points.
  2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of \(\mu(x,y)\) using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
  3. Evaluate the gradient of \(h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3\) at the point \((x,y)=(3,-2)\).

computer Computer Simulation

30 min.

Approximating Functions with Power Series
Static Fields 2022 (8 years)

Taylor series power series approximation

Power Series Sequence (E&M)

Students use prepared Sage code or a prepared Mathematica notebook to plot \(\sin\theta\) simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.

assignment Homework

Flux through a Paraboloid
Static Fields 2022 (4 years)

Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.

face Lecture

5 min.

Energy and Entropy review
Thermal and Statistical Physics 2020 (3 years)

thermodynamics statistical mechanics

This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.

group Small Group Activity

30 min.

Name the experiment (changing entropy)
Energy and Entropy 2021 (2 years)

thermodynamics entropy experiment derivative first law

Students are placed into small groups and asked to create an experimental setup they can use to measure the partial derivative they are given, in which entropy changes.

accessibility_new Kinesthetic

10 min.

Curvilinear Basis Vectors
Static Fields 2022 (6 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).

group Small Group Activity

30 min.

Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (6 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

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.

group Small Group Activity

60 min.

Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (4 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.