Static Fields 2022 (3 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?

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.

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).
Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable.
Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space.
The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.