Series Notation 1

    • assignment Series Notation 2

      assignment Homework

      Series Notation 2

      Power Series Sequence (E&M)

      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      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 Mass of a Slab

      assignment Homework

      Mass of a Slab
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      Determine the total mass of each of the slabs below.

      1. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho=A\pi\sin(\pi z/h). \end{equation}
      2. A square slab of side length \(L\) with thickness \(h\), resting on a table top at \(z=0\), whose mass density is given by \begin{equation} \rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big) \end{equation}
      3. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose surface density is given by \(\sigma=2Ah\).
      4. An infinitesimally thin square sheet of side length \(L\), resting on a table top at \(z=0\), whose mass density is given by \(\rho=2Ah\,\delta(z)\).
      5. What are the dimensions of \(A\)?
      6. Write several sentences comparing your answers to the different cases above.

    • assignment Total Charge

      assignment Homework

      Total Charge
      charge density curvilinear coordinates

      Integration Sequence

      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      For each case below, find the total charge.

      1. A positively charged (dielectric) spherical shell of inner radius \(a\) and outer radius \(b\) with a spherically symmetric internal charge density \begin{equation} \rho(\vec{r})=3\alpha\, e^{(kr)^3} \end{equation}
      2. A positively charged (dielectric) cylindrical shell of inner radius \(a\) and outer radius \(b\) with a cylindrically symmetric internal charge density \begin{equation} \rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks} \end{equation}

    • assignment Sum Shift

      assignment Homework

      Sum Shift
      Central Forces Spring 2021

      In each of the following sums, shift the index \(n\rightarrow n+2\). Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

      1. \begin{equation} \sum_{n=0}^3 n \end{equation}
      2. \begin{equation} \sum_{n=1}^5 e^{in\phi} \end{equation}
      3. \begin{equation} \sum_{n=0}^{\infty} a_n n(n-1)z^{n-2} \end{equation}

    • assignment Paramagnetism

      assignment Homework

      Paramagnetism
      Energy Temperature Paramagnetism Thermal and Statistical Physics Spring 2020 Find the equilibrium value at temperature \(T\) of the fractional magnetization \begin{equation} \frac{\mu_{tot}}{Nm} \equiv \frac{2\langle s\rangle}{N} \end{equation} of a system of \(N\) spins each of magnetic moment \(m\) in a magnetic field \(B\). The spin excess is \(2s\). The energy of this system is given by \begin{align} U &= -\mu_{tot}B \end{align} where \(\mu_{tot}\) is the total magnetization. Take the entropy as the logarithm of the multiplicity \(g(N,s)\) as given in (1.35 in the text): \begin{equation} S(s) \approx k_B\log g(N,0) - k_B\frac{2s^2}{N} \end{equation} for \(|s|\ll N\), where \(s\) is the spin excess, which is related to the magnetization by \(\mu_{tot} = 2sm\). Hint: Show that in this approximation \begin{equation} S(U) = S_0 - k_B\frac{U^2}{2m^2B^2N}, \end{equation} with \(S_0=k_B\log g(N,0)\). Further, show that \(\frac1{kT} = -\frac{U}{m^2B^2N}\), where \(U\) denotes \(\langle U\rangle\), the thermal average energy.
    • group Magnetic Field Due to a Spinning Ring of Charge

      group Small Group Activity

      30 min.

      Magnetic Field Due to a Spinning Ring of Charge
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      magnetic fields current Biot-Savart law vector field symmetry

      Power Series Sequence (E&M)

      Ring Cycle Sequence

      Students work in groups of three to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

      In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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 Magnetic Vector Potential Due to a Spinning Charged Ring

      group Small Group Activity

      30 min.

      Magnetic Vector Potential Due to a Spinning Charged Ring
      AIMS Maxwell AIMS 21 Static Fields Winter 2021

      compare and contrast mathematica magnetic vector potential magnetic fields vector field symmetry

      Power Series Sequence (E&M)

      Ring Cycle Sequence

      Students work in groups of three 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.

    • assignment Nucleus in a Magnetic Field

      assignment Homework

      Nucleus in a Magnetic Field
      Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

      Nuclei of a particular isotope species contained in a crystal have spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\), compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus can be in one of 3 states, two of which have energy \(E=\varepsilon\) and one has energy \(E=0\).

      1. Find the Helmholtz free energy \(F = U-TS\) for a crystal containing \(N\) nuclei which do not interact with each other.

      2. Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

      3. Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

    • assignment Quantum Particle in a 2-D Box

      assignment Homework

      Quantum Particle in a 2-D Box
      Central Forces Spring 2021

      (2 points each)

      You know that the normalized spatial eigenfunctions for a particle in a 1-D box of length \(L\) are \(\sqrt{\frac{2}{L}}\sin{\frac{n\pi x}{L}}\). If you want the eigenfunctions for a particle in a 2-D box, then you just multiply together the eigenfunctions for a 1-D box in each direction. (This is what the separation of variables procedure tells you to do.)

      1. Find the normalized eigenfunctions for a particle in a 2-D box with sides of length \(L_x\) in the \(x\)-direction and length \(L_y\) in the \(y\)-direction.
      2. Find the Hamiltonian for a 2-D box and show that your eigenstates are indeed eigenstates and find a formula for the possible energies
      3. Any sufficiently smooth spatial wave function inside a 2-D box can be expanded in a double sum of the product wave functions, i.e. \begin{equation} \psi(x,y)=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\; \hbox{eigenfunction}_n(x)\;\hbox{eigenfunction}_m(y) \end{equation} Using your expressions from part (a) above, write out all the terms in this sum out to \(n=3\), \(m=3\). Arrange the terms, conventionally, in terms of increasing energy.

        You may find it easier to work in bra/ket notation: \begin{align*} \left|{\psi}\right\rangle &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{n}\right\rangle \left|{m}\right\rangle \\ &=\sum_{n=1}^{\infty}\sum_{m=1}^{\infty}\, c_{nm}\left|{nm}\right\rangle \end{align*}

      4. Find a formula for the \(c_{nm}\)s in part (b). Find the formula first in bra ket notation and then rewrite it in wave function notation.

    • group Superposition States for a Particle on a Ring

      group Small Group Activity

      30 min.

      Superposition States for a Particle on a Ring

      central forces quantum mechanics eigenstates eigenvalues quantum measurements angular momentum hermitian operators probability superposition

      Quantum Ring Sequence

      Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
  • AIMS Maxwell AIMS 21 Static Fields Winter 2021

    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}