Line Sources Using Coulomb's Law

    • 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 Mass Density

      assignment Homework

      Mass Density
      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.
    • 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.
    • assignment Line Sources Using the Gradient

      assignment Homework

      Line Sources Using the Gradient
      Static Fields 2022 (4 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 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?
    • 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 Vector Differential--Curvilinear

      group Small Group Activity

      30 min.

      Vector Differential--Curvilinear
      Vector Calculus II 2022 (7 years)

      vector calculus coordinate systems curvilinear coordinates

      Integration Sequence

      In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

      Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

    • assignment Potential vs. Potential Energy

      assignment Homework

      Potential vs. Potential Energy
      Static Fields 2022 (4 years)

      In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like \(\frac{1}{r}\), so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

      1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
      2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
      3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

    • 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!)

    • group Heat and Temperature of Water Vapor (Remote)

      group Small Group Activity

      5 min.

      Heat and Temperature of Water Vapor (Remote)

      Thermo Heat Capacity Partial Derivatives

      In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
  • Static Fields 2022 (4 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.