Differential Form of Gauss's Law
- assignment Electric Field and Charge
assignment Homework
Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2023 (4 years) Consider the electric field \begin{equation} \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} \end{equation}- Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
- Find a formula for the charge density that creates this electric field.
- Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
- assignment Potential vs. Potential Energy
assignment Homework
Potential vs. Potential Energy
Static Fields 2023 (6 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:
- 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.
- 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.
- 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 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. - assignment Gravitational Field and Mass
assignment Homework
Gravitational Field and Mass
Static Fields 2023 (5 years)The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: \begin{equation} \vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases} \end{equation}
This problem explores the consequences of the divergence theorem for this shell.
- Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: \(r<a\), \(a<r<b\), and \(r>b\).
- Briefly discuss the physical meaning of the divergence in this particular example.
- For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius \(Q\), where \(a<Q<b\). ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
- Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.
- keyboard Electric field for a waffle cone of charge
keyboard Computational Activity
120 min.
Electric field for a waffle cone of charge
Computational Physics Lab II 2022 Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient. - group Gaussian Parameters
group Small Group Activity
10 min.
Gaussian Parameters
Periodic Systems 2022 Students use an applet to explore the role of the parameters \(N\), \(x_o\), and \(\sigma\) in the shape of a Gaussian \begin{equation} f(x)=Ne^{-\frac{(x-x_0)^2}{2\sigma^2}} \end{equation} - assignment_ind Normalization of the Gaussian for Wavefunctions
assignment_ind Small White Board Question
5 min.
Normalization of the Gaussian for Wavefunctions
Periodic Systems 2022 Students find a wavefunction that corresponds to a Gaussian probability density. - keyboard Electrostatic potential and Electric Field of a square of charge
keyboard Computational Activity
120 min.
Electrostatic potential and Electric Field of a square of charge
Computational Physics Lab II 2023 (2 years) Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically. - keyboard Electrostatic potential of spherical shell
keyboard Computational Activity
120 min.
Electrostatic potential of spherical shell
Computational Physics Lab II 2022 Students solve numerically for the potential due to a spherical shell of charge. Although this potential is straightforward to compute using Gauss's Law, it serves as a nice example for numerically integrating in spherical coordinates because the correct answer is easy to recognize. - group Electrostatic Potential Due to a Ring of Charge
group Small Group Activity
30 min.
Electrostatic Potential Due to a Ring of Charge
Static Fields 2023 (8 years)electrostatic potential charge linear charge density taylor series power series scalar field superposition symmetry distance formula
Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(V(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.
-
Static Fields 2023 (6 years)
For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.