Linear Quadrupole (w/ series)

• This problem is used in the following sequences

  1. << Electrostatic Potential Due to a Pair of Charges (with Series) | Power Series Sequence (E&M) | Electrostatic Potential Due to a Ring of Charge >>

• Static Fields 2023 (7 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 \(z=0\).

  1. 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.

  2. 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.

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