*assignment*Linear Quadrupole (w/ series)*assignment*Homework##### Linear Quadrupole (w/ series)

Static Fields 2023 (6 years)Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r'}\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert} \] Electrostatic potentials satisfy the superposition principle.

Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

- A series of charges arranged in this way is called a linear quadrupole. Why?

*group*Equipotential Surfaces*group*Small Group Activity120 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.*keyboard*Electrostatic potential of four point charges*keyboard*Computational Activity120 min.

##### Electrostatic potential of four point charges

Computational Physics Lab II 2023 (2 years)electrostatic potential python

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using`numpy`

and`matplotlib`

.*computer*Using Technology to Visualize Potentials*computer*Mathematica Activity30 min.

##### Using Technology to Visualize Potentials

Static Fields 2023 (6 years)electrostatic potential visualization

Begin by prompting the students to brainstorm different ways to represent a three dimensional scalar field on a 2-D surface (like their paper or a whiteboard). The students use a pre-made*Sage*code or a*Mathematica*worksheet to visualize the electrostatic potential of several distributions of charges. The computer algebra systems demonstrate several different ways of plotting the potential.*group*Electrostatic Potential Due to a Pair of Charges (without Series)*group*Small Group Activity30 min.

##### Electrostatic Potential Due to a Pair of Charges (without Series)

Static Fields 2023 (4 years) Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.*group*Work By An Electric Field (Contour Map)*group*Small Group Activity30 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*Number of Paths*group*Small Group Activity30 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.*format_list_numbered*Ring Cycle Sequence*format_list_numbered*Sequence##### Ring Cycle Sequence

Students calculate electrostatic fields (\(V\), \(\vec{E}\)) and magnetostatic fields (\(\vec{A}\), \(\vec{B}\)) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.*assignment*Nucleus in a Magnetic Field*assignment*Homework##### Nucleus in a Magnetic Field

Energy and Entropy 2021 (2 years)Nuclei of a particular isotope species contained in a crystal have spin \(I=1\), and thus, \(m = \{+1,0,-1\}\). The interaction between the nuclear quadrupole moment and the gradient of the crystalline electric field produces a situation where the nucleus has the same energy, \(E=\varepsilon\), in the state \(m=+1\) and the state \(m=-1\), compared with an energy \(E=0\) in the state \(m=0\), i.e. each nucleus can be in one of 3 states, two of which have energy \(E=\varepsilon\) and one has energy \(E=0\).

Find the Helmholtz free energy \(F = U-TS\) for a crystal containing \(N\) nuclei which do not interact with each other.

Find an expression for the entropy as a function of temperature for this system. (Hint: use results of part a.)

- Indicate what your results predict for the entropy at the extremes of very high temperature and very low temperature.

*accessibility_new*Acting Out Charge Densities*accessibility_new*Kinesthetic10 min.

##### Acting Out Charge Densities

Static Fields 2023 (6 years)density charge density mass density linear density uniform idealization

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.-
Static Fields 2023 (4 years)
Consider a collection of three charges arranged in a line along the
\(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).
Find the electrostatic potential at a point \(\vec{r}\) on the \(x\)-axis at a distance \(x\) from the center of the quadrupole.

A series of charges arranged in this way is called a linear quadrupole. Why?