Power Series Coefficients 2

    • assignment Power Series Coefficients 3

      assignment Homework

      Power Series Coefficients 3
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 Use the formula for a Taylor series: \[f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n\] to find the first three non-zero terms of a series expansion for \(f(z)=\cos(kz)\) around \(z=2\).
    • assignment Series Notation 2

      assignment Homework

      Series Notation 2

      Power Series Sequence (E&M)

      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      Write (a good guess for) the following series using sigma \(\left(\sum\right)\) notation. (If you only know a few terms of a series, you don't know for sure how the series continues.)

      1. \[1 - 2\,\theta^2 + 4\,\theta^4 - 8\,\theta^6 +\,\dots\]

      2. \[\frac14 - \frac19 + \frac{1}{16} - \frac{1}{25}+\,\dots\]

    • assignment Series Notation 1

      assignment Homework

      Series Notation 1

      Power Series Sequence (E&M)

      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      Write out the first four nonzero terms in the series:

      1. \[\sum\limits_{n=0}^\infty \frac{1}{n!}\]

      2. \[\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}\]
      3. \begin{equation} \sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}} \end{equation}

    • assignment Memorize Power Series

      assignment Homework

      Memorize Power Series

      Power Series Sequence (E&M)

      Static Fields Winter 2021

      Look up and memorize the power series to fourth order for \(e^z\), \(\sin z\), \(\cos z\), \((1+z)^p\) and \(\ln(1+z)\). For what values of \(z\) do these series converge?

    • assignment Series Convergence

      assignment Homework

      Series Convergence

      Power Series Sequence (E&M)

      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      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 Approximating Functions with Power Series

      computer Computer Simulation

      30 min.

      Approximating Functions with Power Series
      Theoretical Mechanics Fall 2020 Theoretical Mechanics Fall 2021 AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Central Forces Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      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 the power series expansion in a previous activity, Calculating Coefficients for a Power Series.
    • assignment Linear Quadrupole (w/ series)

      assignment Homework

      Linear Quadrupole (w/ series)

      Power Series Sequence (E&M)

      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

      1. Find the electrostatic potential at a point \(P\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole.
      2. 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.
      3. A series of charges arranged in this way is called a linear quadrupole. Why?

    • assignment Line Sources Using Coulomb's Law

      assignment Homework

      Line Sources Using Coulomb's Law
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022
      1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance \(s\) straight out from the midpoint, starting from Coulomb's Law.
      2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.
    • group Electric Field Due to a Ring of Charge

      group Small Group Activity

      30 min.

      Electric Field Due to a Ring of Charge
      AIMS Maxwell Fall 21 AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      coulomb's law electric field charge ring symmetry integral power series superposition

      Power Series Sequence (E&M)

      Ring Cycle Sequence

      Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

      In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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 Electrostatic Potential Due to a Pair of Charges (with Series)

      group Small Group Activity

      60 min.

      Electrostatic Potential Due to a Pair of Charges (with Series)
      AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022

      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.
  • AIMS Maxwell Fall 21 AIMS Maxwell Spring 2021 Static Fields Winter 2021 Static Fields Winter 2022 Use the formula for a Taylor series: \[f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n\] to find the first three non-zero terms of a series expansion for \(f(z)=e^{-kz}\) around \(z=3\).