Symmetry Arguments for Gauss's Law
- accessibility_new Time Dilation Light Clock Skit
accessibility_new Kinesthetic
5 min.
Time Dilation Light Clock Skit
Special Relativity Time Dilation Light Clock Kinesthetic Activity
Students act out the classic light clock scenario for deriving time dilation. - group Systems of Equations Compare and Contrast
- group Electric Field Due to a Ring of Charge
group Small Group Activity
30 min.
Electric Field Due to a Ring of Charge
Static Fields 2022 (7 years)coulomb's law electric field charge ring symmetry integral power series superposition
Students work in groups of three to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.
In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.
- 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 \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 Gravitational Field and Mass
assignment Homework
Gravitational Field and Mass
Static Fields 2022 (4 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.
- group Vector Integrals (Contour Map)
- group Equipotential Surfaces
group Small Group Activity
120 min.
Equipotential Surfaces
Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials. - group Work By An Electric Field (Contour Map)
group Small Group Activity
30 min.
Work By An Electric Field (Contour Map)
Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential. - group Electric Field of Two Charged Plates
group Small Group Activity
30 min.
Electric Field of Two Charged Plates
- Students need to understand that the surface represents the electric potential in the center of a parallel plate capacitor. Try doing the activity Electric Potential of Two Charged Plates before this activity.
- Students should know that
- objects with like charge repel and opposite charge attract,
- object tend to move toward lower energy configurations
- The potential energy of a charged particle is related to its charge: \(U=qV\)
- The force on a charged particle is related to its charge: \(\vec{F}=q\vec{E}\)
- group Charged Sphere
group Small Group Activity
30 min.
Charged Sphere
Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field. -
Static Fields 2022 (4 years)
Instructions for 2022: You will need to complete this assignment in a 15 minute appointment on Zoom or in person with one of the members of the teaching team between 1/21 and 10 pm on 1/26. Here is a link to a sign-up page.
You are required to watch a sample video for how to make symmetry arguments here. As demonstrated in the video you should bring with you to the meeting a cylinder, an observer, and a vector.
Use good symmetry arguments to find the possible direction for the electric field due to a charged wire. Also, use good symmetry arguments to find the possible functional dependence of the electric field due to a charged wire. Rather than writing this up to turn in, you should find a member of the teaching team and make the arguments to them verbally.