Electric Field of a Dipole at a General Point

The axial result of Section 1.9 and the equatorial result of Section 1.10 are, in fact, two special cases of a single general formula for the field of a short dipole at a point $P$ whose position vector (measured from the dipole's centre $O$) makes an angle $\theta$ with the dipole axis.

Resolving the field at such a general point into a component along the radial direction (the line $OP$ extended) and a component along the direction perpendicular to $OP$ (tangential), the two components work out to be

Combining these two perpendicular components by the Pythagorean rule gives the magnitude of the total field:

(using $\sin^2\theta=1-\cos^2\theta$, so $4\cos^2\theta+\sin^2\theta = 4\cos^2\theta+1-\cos^2\theta = 1+3\cos^2\theta$).

Checking the two special cases already derived:

At $\theta=0^\circ$ (a point on the dipole axis, Section 1.9): $\cos\theta=1$, so $E=(kp/r^3)\sqrt{1+3}=(kp/r^3)\times2 = 2kp/r^3$ -- exactly matching $E_{\text{axial}}$.

At $\theta=90^\circ$ (a point on the equatorial line, Section 1.10): $\cos\theta=0$, so $E=(kp/r^3)\sqrt{1+0}=kp/r^3$ -- exactly matching $E_{\text{eq}}$. …