Mass Density

    • assignment Gauss's Law for a Rod inside a Cube

      assignment Homework

      Gauss's Law for a Rod inside a Cube
      Static Fields 2022 (3 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(x,y\)-plane. The charge density \(\lambda_0\) is constant. Find the total flux of the electric field through a closed cubical surface with sides of length \(3L\) centered at the origin.
    • 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
      Static Fields 2022 (5 years)

      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 Electric Field from a Rod

      assignment Homework

      Electric Field from a Rod
      Static Fields 2022 (4 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(xy\)-plane. The charge density \(\lambda\) is constant. Find the electric field at the point \((0,0,2L)\).
    • assignment Line Sources Using Coulomb's Law

      assignment Homework

      Line Sources Using Coulomb's Law
      Static Fields 2022 (5 years)
      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.
    • assignment Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient

      Gradient Sequence

      Static Fields 2022 (5 years)
      1. Find the electric field around an infinite, uniformly charged, straight wire, starting from the following expression for the electrostatic potential: \begin{equation} V(\vec r)=\frac{2\lambda}{4\pi\epsilon_0}\, \ln\left( \frac{ s_0}{s} \right) \end{equation}

    • assignment Helix

      assignment Homework

      Helix

      Integration Sequence

      Static Fields 2022 (5 years)

      A helix with 17 turns has height \(H\) and radius \(R\). Charge is distributed on the helix so that the charge density increases like (i.e. proportional to) the square of the distance up the helix. At the bottom of the helix the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the top of the helix, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the helix?

    • accessibility_new Acting Out Charge Densities

      accessibility_new Kinesthetic

      10 min.

      Acting Out Charge Densities
      Static Fields 2022 (5 years)

      density charge density mass density linear density uniform idealization

      Ring Cycle Sequence

      Integration Sequence

      Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.
    • assignment Gibbs sum for a two level system

      assignment Homework

      Gibbs sum for a two level system
      Gibbs sum Microstate Thermal average energy Thermal and Statistical Physics 2020
      1. Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.

      2. Solve for the thermal average occupancy of the system in terms of \(\lambda\).

      3. Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

      4. Find an expression for the thermal average energy of the system.

      5. Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.

    • assignment Extensive Internal Energy

      assignment Homework

      Extensive Internal Energy
      Energy and Entropy 2021 (2 years)

      Consider a system which has an internal energy \(U\) defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal energy is an extensive quantity. What constraint does this place on the values \(\alpha\) and \(\beta\) may have?

    • group Total Charge

      group Small Group Activity

      30 min.

      Total Charge
      Static Fields 2022 (5 years)

      charge charge density multiple integral scalar field coordinate systems differential elements curvilinear coordinates

      Integration Sequence

      In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
  • Static Fields 2022 (3 years) Consider a rod of length \(L\) lying on the \(z\)-axis. Find an algebraic expression for the mass density of the rod if the mass density at \(z=0\) is \(\lambda_0\) and at \(z=L\) is \(7\lambda_0\) and you know that the mass density increases
    • linearly;
    • like the square of the distance along the rod;
    • exponentially.