Consider a dipole of strength D, oriented along the x-axis and located at the point x = ε/2, y = 0, z = 0, and a dipole of strength -D, oriented along the x-axis at the point x = - ε /2, y = 0, z = 0. Write the expression for the potential for these two dipoles. Take the limit as ε →0, εD = E = fixed, and show that the limiting potential is given by
u= -E/4π3(3cos2 θ - 1)
This configuration is a quadrupole. Show that u satisfies
?u = δ"(x) δ(y) δ(z).