Line Sources Using Coulomb's Law

    • assignment Electric Field from a Rod

      assignment Homework

      Electric Field from a Rod
      Static Fields 2022 (5 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 Mass Density

      assignment Homework

      Mass Density
      Static Fields 2022 (4 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.
    • assignment Gauss's Law for a Rod inside a Cube

      assignment Homework

      Gauss's Law for a Rod inside a Cube
      Static Fields 2022 (4 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.
    • assignment Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient

      Gradient Sequence

      Static Fields 2022 (6 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 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?

    • assignment Boltzmann probabilities

      assignment Homework

      Boltzmann probabilities
      Energy Entropy Boltzmann probabilities Thermal and Statistical Physics 2020 (3 years) Consider a three-state system with energies \((-\epsilon,0,\epsilon)\).
      1. At infinite temperature, what are the probabilities of the three states being occupied? What is the internal energy \(U\)? What is the entropy \(S\)?
      2. At very low temperature, what are the three probabilities?
      3. What are the three probabilities at zero temperature? What is the internal energy \(U\)? What is the entropy \(S\)?
      4. What happens to the probabilities if you allow the temperature to be negative?
    • face Wavelength of peak intensity

      face Lecture

      5 min.

      Wavelength of peak intensity
      Contemporary Challenges 2022 (3 years)

      Wein's displacement law blackbody radiation

      This very short lecture introduces Wein's displacement law.
    • computer Blackbody PhET

      computer Computer Simulation

      30 min.

      Blackbody PhET
      Contemporary Challenges 2022 (4 years)

      blackbody

      Students use a PhET to explore properties of the Planck distribution.
    • assignment Electric Field of a Finite Line

      assignment Homework

      Electric Field of a Finite Line

      Consider the finite line with a uniform charge density from class.

      1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
      2. Perform the integral to find the \(z\)-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the \(s\)-component as well!)

  • Static Fields 2022 (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.