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 \(P\) in the \(xy\)-plane at a
distance \(s\) from the center of the quadrupole.

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?

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.

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 demonstrates several different ways of plotting the potential.

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.

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.

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.