Yukawa

    • assignment Helix

      assignment Homework

      Helix

      Integration Sequence

      Static Fields 2023 (6 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?

    • assignment Undo Formulas for Reduced Mass (Geometry)

      assignment Homework

      Undo Formulas for Reduced Mass (Geometry)
      Central Forces 2023 (3 years)

      The figure below shows the position vector \(\vec r\) and the orbit of a “fictitious” reduced mass \(\mu\).

      1. Suppose \(m_1=m_2\), Sketch the position vectors and orbits for \(m_1\) and \(m_2\) corresponding to \(\vec{r}\). Describe a common physics example of central force motion for which \(m_1=m_2\).
      2. Repeat, for \(m_2>m_1\).

    • assignment Charge on a Spiral

      assignment Homework

      Charge on a Spiral
      Static Fields 2023 (2 years) A charged spiral in the \(x,y\)-plane has 6 turns from the origin out to a maximum radius \(R\) , with \(\phi\) increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is \(0~\frac{\textrm{C}}{\textrm{m}}\). At the end of the spiral, the linear charge density is \(13~\frac{\textrm{C}}{\textrm{m}}\). What is the total charge on the spiral?
    • group Box Sliding Down Frictionless Wedge

      group Small Group Activity

      120 min.

      Box Sliding Down Frictionless Wedge
      Theoretical Mechanics (4 years)

      Lagrangian Mechanics Generalized Coordinates Special Cases

      Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
    • assignment Find Force Law: Logarithmic Spiral Orbit

      assignment Homework

      Find Force Law: Logarithmic Spiral Orbit
      Central Forces 2023 (3 years)

      In science fiction movies, characters often talk about a spaceship “spiralling in” right before it hits the planet. But all orbits in a \(1/r^2\) force are conic sections, not spirals. This spiralling in happens because the spaceship hits atmosphere and the drag from the atmosphere changes the shape of the orbit. But, in an alternate universe, we might have other force laws.

      Find the force law for a mass \(\mu\), under the influence of a central-force field, that moves in a logarithmic spiral orbit given by \(r = ke^{\alpha \phi}\), where \(k\) and \(\alpha\) are constants.

    • face Gibbs entropy approach

      face Lecture

      120 min.

      Gibbs entropy approach
      Thermal and Statistical Physics 2020

      Gibbs entropy information theory probability statistical mechanics

      These lecture notes for the first week of Thermal and Statistical Physics include a couple of small group activities in which students work with the Gibbs formulation of the entropy.
    • 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?

    • assignment Magnetic Field and Current

      assignment Homework

      Magnetic Field and Current
      Static Fields 2023 (4 years) Consider the magnetic field \[ \vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases} \]
      1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
      2. Find a formula for the current density that creates this magnetic field.
      3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
    • assignment Total Charge

      assignment Homework

      Total Charge
      charge density curvilinear coordinates

      Integration Sequence

      Static Fields 2023 (6 years)

      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 Line Sources Using Coulomb's Law

      assignment Homework

      Line Sources Using Coulomb's Law
      Static Fields 2023 (6 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.
  • Central Forces 2023 (3 years)

    In a solid, a free electron doesn't see” a bare nuclear charge since the nucleus is surrounded by a cloud of other electrons. The nucleus will look like the Coulomb potential close-up, but be screened” from far away. A common model for such problems is described by the Yukawa or screened potential: \begin{equation} U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}} \end{equation}

    1. Graph the potential, with and without the exponential term. Describe how the Yukawa potential approximates the “real” situation. In particular, describe the role of the parameter \(\alpha\).
    2. Draw the effective potential for the two choices \(\alpha=10\) and \(\alpha=0.1\) with \(k=1\) and \(\ell=1\). For which value(s) of \(\alpha\) is there the possibility of stable circular orbits?

  • Media & Figures
    • figures/cfyukawa.svg
    • figures/cfyukawaeffective.svg