## Cube Charge

• This problem is used in the following sequences
• charge density
• computer Visualizing Flux through a Cube

computer Computer Simulation

30 min.

##### Visualizing Flux through a Cube
Static Fields 2022 (4 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 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 Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2022 (4 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 $$\rho(\vec{r})=3\alpha\, e^{(kr)^3}$$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

• assignment Cross Triangle

assignment Homework

##### Cross Triangle
Static Fields 2022 (4 years)

Use the cross product to find the components of the unit vector $\mathbf{\boldsymbol{\hat n}}$ perpendicular to the plane shown in the figure below, i.e.  the plane joining the points $\{(1,0,0),(0,1,0),(0,0,1)\}$.

• assignment Tetrahedron

assignment Homework

##### Tetrahedron
Static Fields 2022 (4 years)

Using a dot product, find the angle between any two line segments that join the center of a regular tetrahedron to its vertices. Hint: Think of the vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates (0,0,0), (1,1,0), (1,0,1) and (0,1,1)---you may need to build a model and play with it to see how this works!)

• biotech Microwave oven Ice Calorimetry Lab

biotech Experiment

60 min.

##### Microwave oven Ice Calorimetry Lab
Energy and Entropy 2021 (2 years)

In this remote-friendly activity, students use a microwave oven (and optionally a thermometer) to measure the latent heat of melting for water (and optionally the heat capacity). From these they compute changes in entropy. See also Ice Calorimetry Lab.
• assignment Divergence through a Prism

assignment Homework

##### Divergence through a Prism
Static Fields 2022 (4 years)

Consider the vector field $\vec F=(x+2)\hat{x} +(z+2)\hat{z}$.

1. Calculate the divergence of $\vec F$.
2. In which direction does the vector field $\vec F$ point on the plane $z=x$? What is the value of $\vec F\cdot \hat n$ on this plane where $\hat n$ is the unit normal to the plane?
3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

• 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.
• assignment Reduced Mass

assignment Homework

##### Reduced Mass
Central Forces 2021

Using your favorite graphing package, make a plot of the reduced mass $\mu$ as a function of $m_1$ and $m_2$. What about the shape of this graph tells you something about the physical world that you would like to remember. You should be able to find at least three things.

• assignment Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• Static Fields 2022 (4 years)
1. Charge is distributed throughout the volume of a dielectric cube with charge density $\rho=\beta z^2$, where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
2. In a new physical situation: Charge is distributed on the surface of a cube with charge density $\sigma=\alpha z$ where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge on the cube? Don't forget about the top and bottom of the cube.