Flux through a Paraboloid

• group Scalar Surface and Volume Elements

  group Small Group Activity

  30 min.

  Scalar Surface and Volume Elements

  Static Fields 2023 (7 years)

  Integration Sequence

  Students use known algebraic expressions for length elements \(d\ell\) to determine all simple scalar area \(dA\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

  This activity is identical to Vector Surface and Volume Elements except uses a scalar approach to find surface, and volume elements.

• assignment Find Area/Volume from $d\vec{r}$

  assignment Homework

  Find Area/Volume from \(d\vec{r}\)

  Static Fields 2023 (5 years)

  Start with \(d\vec{r}\) in rectangular, cylindrical, and spherical coordinates. Use these expressions to write the scalar area elements \(dA\) (for different coordinate equals constant surfaces) and the volume element \(d\tau\). It might help you to think of the following surfaces: The various sides of a rectangular box, a finite cylinder with a top and a bottom, a half cylinder, and a hemisphere with both a curved and a flat side, and a cone.

  1. Rectangular: \begin{align} dA&=\\ d\tau&= \end{align}
  2. Cylindrical: \begin{align} dA&=\\ d\tau&= \end{align}
  3. Spherical: \begin{align} dA&=\\ d\tau&= \end{align}

• group Visualizing Plane Waves

  group Small Group Activity

  60 min.

  Visualizing Plane Waves

  Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

  Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.

• assignment Cone Surface

  assignment Homework

  Cone Surface

  Static Fields 2023 (6 years)

  • Find \(dA\) on the surface of an (open) cone in both cylindrical and spherical coordinates. Hint: Be smart about how you coordinatize the cone.
  • Using integration, find the surface area of an (open) cone with height \(H\) and radius \(R\). Do this problem in both cylindrical and spherical coordinates.

• face Ideal Gas

  face Lecture

  120 min.

  Ideal Gas

  Thermal and Statistical Physics 2020

  ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics

  These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.
• group Generalized Leibniz Notation

  group Small Group Activity

  10 min.

  Generalized Leibniz Notation

  Static Fields 2023 (5 years) This short small group activity introduces students to the Leibniz notation used for partial derivatives in thermodynamics; unlike standard Leibniz notation, this notation explicitly specifies constant variables. Students are guided in linking the variables from a contextless Leibniz-notation partial derivative to their proper variable categories.
• assignment Zapping With d 1

  assignment Homework

  Zapping With d 1

  Energy and Entropy 2021 (2 years)

  Find the differential of each of the following expressions; zap each of the following with \(d\):

  1. \[f=3x-5z^2+2xy\]

  2. \[g=\frac{c^{1/2}b}{a^2}\]

  3. \[h=\sin^2(\omega t)\]

  4. \[j=a^x\]

  5. \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]

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

  assignment Homework

  Gauss's Law for a Rod inside a Cube

  Static Fields 2023 (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.
• face Boltzmann probabilities and Helmholtz

  face Lecture

  120 min.

  Boltzmann probabilities and Helmholtz

  Thermal and Statistical Physics 2020

  ideal gas entropy canonical ensemble Boltzmann probability Helmholtz free energy statistical mechanics

  These notes, from the third week of Thermal and Statistical Physics cover the canonical ensemble and Helmholtz free energy. They include a number of small group activities.
• assignment Current from a Spinning Cylinder

  assignment Homework

  Current from a Spinning Cylinder

  A solid cylinder with radius \(R\) and height \(H\) has its base on the \(x,y\)-plane and is symmetric around the \(z\)-axis. There is a fixed volume charge density on the cylinder \(\rho=\alpha z\). If the cylinder is spinning with period \(T\):

  1. Find the volume current density.
  2. Find the total current.
• Static Fields 2021 (5 years)

  Find the upward pointing flux of the electric field \(\vec E =E_0\, z\, \hat z\) through the part of the surface \(z=-3 s^2 +12\) (cylindrical coordinates) that sits above the \((x, y)\)--plane.