Electric Field of a Dipole on Its Axis

Consider a dipole made of charges $+q$ at point $A$ and $-q$ at point $B$, a distance $2a$ apart, with centre $O$, and consider a point $P$ lying on the extension of the dipole axis beyond the positive charge, at distance $r$ from the centre $O$ (so that $P$ is at distance $r-a$ from $+q$ and at distance $r+a$ from $-q$).

The field at $P$ due to the positive charge $+q$ points away from $+q$, i.e. along the axis away from $O$, with magnitude

The field at $P$ due to the negative charge $-q$ points toward $-q$, i.e. along the axis back toward $O$, with magnitude

Since both fields lie exactly along the same straight line (the dipole axis), the net field is simply their difference, and because $P$ is nearer to $+q$ than to $-q$, $E_+ > E_-$, so the net field points in the direction of $E_+$, i.e. away from $O$ along the axis:

For a short dipole, $r\gg a$, so $a^2$ can be dropped compared to $r^2$ in the denominator, giving $(r^2-a^2)^2\approx r^4$:

$E_{\text{axial}} \approx \frac{4kqar}{r^4} = \frac{2k(q\cdot2a)}{r^3} = \frac{2kp}{r^3}$ …