Adipole is oriented along the x axis. the dipole moment is p (= qs). (assume the center of the dipole is located at the origin with positive charge to the right and negative charge to the left.)

(b) what are the approximate values of v at the locations in part (a) if these locations are far from the dipole? (use the following as necessary: p, x, ε0, and r. it is convenient to use the "binomial expansion" that you may have learned in calculus, that (1 + ε)n ≈ 1 + nε if ε ≪ 1. note that n can be negative.)

v(x,0,0) ≈

v(0,y,0) ≈

(c) using the approximate results of part (b), calculate the gradient of the potential along the x axis. (use the following as necessary: p, ε0 and x.)

∂v
∂x
=
show that the negative gradient is equal to the x component ex of the electric field.

(d) along the y axis, dv/dy = 0. why isn't this equal to the magnitude of the electric field e along the y axis?