assignment Homework

Divergence
Static Fields 2022 (5 years)

Shown above is a two-dimensional vector field.

Determine whether the divergence at point A and at point C is positive, negative, or zero.

assignment Homework

Divergence through a Prism
Static Fields 2022 (5 years)

Consider the vector field \(\vec F=(x+2)\hat{x} +(z+2)\hat{z}\).

  1. Calculate the divergence of \(\vec F\).
  2. In which direction does the vector field \(\vec F\) point on the plane \(z=x\)? What is the value of \(\vec F\cdot \hat n\) on this plane where \(\hat n\) is the unit normal to the plane?
  3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

group Small Group Activity

30 min.

Visualization of Divergence
Vector Calculus II 2022 (8 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.

assignment Homework

Divergence Practice including Curvilinear Coordinates

Calculate the divergence of each of the following vector fields. You may look up the formulas for divergence in curvilinear coordinates.

  1. \begin{equation} \hat{F}=z^2\,\hat{x} + x^2 \,\hat{y} -y^2 \,\hat{z} \end{equation}
  2. \begin{equation} \hat{G} = e^{-x} \,\hat{x} + e^{-y} \,\hat{y} +e^{-z} \,\hat{z} \end{equation}
  3. \begin{equation} \hat{H} = yz\,\hat{x} + zx\,\hat{y} + xy\,\hat{z} \end{equation}
  4. \begin{equation} \hat{I} = x^2\,\hat{x} + z^2\,\hat{y} + y^2\,\hat{z} \end{equation}
  5. \begin{equation} \hat{J} = xy\,\hat{x} + xz\,\hat{y} + yz\,\hat{z} \end{equation}
  6. \begin{equation} \hat{K} = s^2\,\hat{s} \end{equation}
  7. \begin{equation} \hat{L} = r^3\,\hat{\phi} \end{equation}

assignment Homework

Gravitational Field and Mass
Static Fields 2022 (4 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

Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2022 (3 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.

assignment Homework

Differential Form of Gauss's Law
Static Fields 2022 (5 years)

For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

assignment Homework

Curl
Static Fields 2022 (5 years)

Shown above is a two-dimensional cross-section of a vector field. All the parallel cross-sections of this field look exactly the same. Determine the direction of the curl at points A, B, and C.

assignment Homework

Flux through a Paraboloid
Static Fields 2022 (5 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.

assignment Homework

Series Convergence

Power Series Sequence (E&M)

Static Fields 2022 (5 years)

Recall that, if you take an infinite number of terms, the series for \(\sin z\) and the function itself \(f(z)=\sin z\) are equivalent representations of the same thing for all real numbers \(z\), (in fact, for all complex numbers \(z\)). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of \(z\). The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”

Find the power series for the function \(f(z)=\frac{1}{1+z^2}\). Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!

Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”

computer Computer Simulation

30 min.

Approximating Functions with Power Series
Theoretical Mechanics (12 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.

face Lecture

120 min.

Fermi and Bose gases
Thermal and Statistical Physics 2020

Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.

face Lecture

120 min.

Phase transformations
Thermal and Statistical Physics 2020

phase transformation Clausius-Clapeyron mean field theory thermodynamics

These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.