Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.

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.

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 examine a plastic "surface" graph of the electric potential due to two changes plates (near the center of the plates) and explore the properties of the electric potential.

In a solid, a free electron doesn't see” a bare nuclear charge
since the nucleus is surrounded by a cloud of other electrons. The
nucleus will look like the Coulomb potential close-up, but be
screened” from far away. A common model for such problems is
described by the Yukawa or screened potential:
\begin{equation}
U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}}
\end{equation}

Graph the potential, with and without the exponential term.
Describe how the Yukawa potential approximates the “real”
situation. In particular, describe the role of the parameter
\(\alpha\).

Draw the effective potential for the two choices \(\alpha=10\) and
\(\alpha=0.1\) with \(k=1\) and \(\ell=1\). For which value(s) of
\(\alpha\) is there the possibility of stable circular orbits?