assignment Homework

Divergence
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Shown above is a two-dimensional vector field.

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

group Small Group Activity

30 min.

Visualization of Divergence
AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021 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 through a Prism
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

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
AIMS Maxwell AIMS 21

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 value 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. 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. 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 AIMS Maxwell AIMS 21 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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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

Flux through a Paraboloid
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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)

AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

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.

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
AIMS Maxwell AIMS 21 Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021 Central Forces Spring 2021 Static Fields Winter 2021

Taylor series power series approximation

Power Series Sequence (E&M)

Students use 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 a power series expansion in a previous activity, Calculating Coefficients for a Power Series.