Electric Potential Due to an Electric Dipole

An electric dipole consists of two equal and opposite point charges, $+q$ and $-q$, separated by a small distance $2a$, characterised by its dipole moment $\vec{p}$, a vector of magnitude $p = q(2a)$ directed from the negative charge toward the positive charge. Because a dipole's net charge is exactly zero, its potential pattern is genuinely different in shape from that of a single point charge, and working it out is a natural first application of the scalar superposition principle.

Consider a point $P$ at distance $r$ from the dipole's centre $O$, with $r$ much larger than the charge separation $2a$ (the standard "far-field" approximation used throughout this topic), and let $\theta$ be the angle between the line $OP$ and the dipole axis (the direction of $\vec{p}$). Let $r_1$ and $r_2$ be the exact distances from $P$ to $+q$ and $-q$ respectively. By superposition, the net potential at $P$ is simply the algebraic sum of the two point-charge potentials:

For $r \gg a$, geometry gives the standard far-field approximations $r_2 - r_1 \approx 2a\cos\theta$ and $r_1 r_2 \approx r^2$ (both charges are, to this approximation, at distance $r$ from $P$, and only the small DIFFERENCE $r_2 - r_1$, not the individual distances themselves, needs the more careful geometric estimate). Substituting both approximations,

$V(P) \approx \frac{kq(2a\cos\theta)}{r^2} = \frac{kp\cos\theta}{r^2}$ …