assignment Homework

Total Current, Square Cross-Section

Integration Sequence

Static Fields 2022 (6 years)
  1. Current \(I\) flows down a wire with square cross-section. The length of the square side is \(L\). If the current is uniformly distributed over the entire area, find the current density .
  2. If the current is uniformly distributed over the outer surface only, find the current density .

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.

computer Computer Simulation

30 min.

Blackbody PhET
Contemporary Challenges 2022 (4 years)

blackbody

Students use a PhET to explore properties of the Planck distribution.

group Small Group Activity

5 min.

Acting Out Flux
Static Fields 2022 (4 years)

flux electrostatics vector fields

Students hold rulers and meter sticks to represent a vector field. The instructor holds a hula hoop to represent a small area element. Students are asked to describe the flux of the vector field through the area element.

assignment Homework

Total Current, Circular Cross Section

Integration Sequence

Static Fields 2022 (5 years)

A current \(I\) flows down a cylindrical wire of radius \(R\).

  1. If it is uniformly distributed over the surface, give a formula for the surface current density \(\vec K\).
  2. If it is distributed in such a way that the volume current density, \(|\vec J|\), is inversely proportional to the distance from the axis, give a formula for \(\vec J\).

group Small Group Activity

30 min.

Ideal Gas Model

Ideal Gas surfaces thermo

Students consider whether the thermo surfaces reflect the properties of an ideal gas.

group Small Group Activity

30 min.

Directional Derivatives
Vector Calculus I 2022

Directional derivatives

Gradient Sequence

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

group Small Group Activity

30 min.

Scalar Surface and Volume Elements
Static Fields 2022 (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 Homework

Divergence
Static Fields 2022 (6 years)

Shown above is a two-dimensional vector field.

Determine whether the divergence at point A and at point C is positive, negative, or zero.

group Small Group Activity

30 min.

Number of Paths

E&M Conservative Fields Surfaces

Student discuss how many paths can be found on a map of the vector fields \(\vec{F}\) for which the integral \(\int \vec{F}\cdot d\vec{r}\) is positive, negative, or zero. \(\vec{F}\) is conservative. They do a similar activity for the vector field \(\vec{G}\) which is not conservative.

computer Computer Simulation

30 min.

Visualizing Flux through a Cube
Static Fields 2022 (6 years) Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.

assignment Homework

Find Area/Volume from \(d\vec{r}\)
Static Fields 2022 (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}

assignment Homework

Surface temperature of the Earth
Temperature Radiation Thermal and Statistical Physics 2020 Calculate the temperature of the surface of the Earth on the assumption that as a black body in thermal equilibrium it reradiates as much thermal radiation as it receives from the Sun. Assume also that the surface of the Earth is a constant temperature over the day-night cycle. Use the sun's surface temperature \(T_{\odot}=5800\text{K}\); and the sun's radius \(R_{\odot}=7\times 10^{10}\text{cm}\); and the Earth-Sun distance of \(1.5\times 10^{13}\text{cm}\).

group Small Group Activity

30 min.

Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.

assignment Homework

Sphere in Cylindrical Coordinates
Static Fields 2022 (4 years) Find the surface area of a sphere using cylindrical coordinates.

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 Homework

Flux through a Paraboloid
Static Fields 2022 (6 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.

group Small Group Activity

30 min.

Covariation in Thermal Systems

Thermo Multivariable Functions

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.

group Small Group Activity

30 min.

Gravitational Force

Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.

group Small Group Activity

30 min.

Vector Surface and Volume Elements
Static Fields 2022 (4 years)

Integration Sequence

Students use known algebraic expressions for vector line elements \(d\vec{r}\) to determine all simple vector area \(d\vec{A}\) and volume elements \(d\tau\) in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.