A charge configuration consists of three point charges

Chapter , Problem 34

(choose chapter or problem)

A charge configuration consists of three point charges located on the z axis (Figure 23-33). One has a charge equal to -2q, and is located at the origin. The other two each have a charge equal to +q, one is located at z=+L and the other is located at z=-L. This charge configuration can be modeled as two dipoles: one centered at z=+L/2 and with a dipole moment in the +z direction, the other centered at z=-L/2 and with a dipole moment in the -z direction. Each of these dipoles has a dipole moment that has a magnitude equal to qL. Two dipoles arranged in this fashion form a linear electric quadrupole. (There are other geometrical arrangements of dipoles that create quadrupoles but they are not linear.)

(a) Using the result from Problem 33, show that at large distances from the quadrupole (i.e., for \(r \gg L\) ), the electric potential is given by \(V_{\text {quad }}(r, \theta)=2 k B \cos ^2 \theta / r^3\), where \(B=q L^2\). ( B is the magnitude of the quadrupole moment of the charge configuration.)

(b) Show that on the positive z axis, this potential gives an electric field (for \(z \gg L\) ) of \(\vec{E}=\left(6 k B / z^4\right) \hat{k}\).

(c) Show you get the result of Part (b) by adding the electric fields from the three point charges.