Electric field due to a small dipole at a general point

 

Electric field due to a small dipole at a general point so general point means it is not in the axial line it is not in the equatorial line so earlier we have done derivations for axial line and equatorial line so now this is for the general point so general point means it can be anywhere so that means you are given a distance of r and the angle given $\theta$ with respect to the dipole, so the method or the principle to arrive at the solution is very simple. Dipole moment is resolved into two components $p\,cos\,\theta$ along the r line and $p\,sin\,\theta$ perpendicular to r direction. Then it becomes a very easy problem.

\[ a \ll r \] \[ E_{\text{axial}}=\frac{k\,2P\cos\theta}{r^3} \] \[ E_{\text{equatorial}}=\frac{k\,P\sin\theta}{r^3} \] \[ E_{\text{resultant}} = \sqrt{E_{\text{axial}}^{\,2}+E_{\text{equatorial}}^{\,2}} \] \[ = \frac{kP}{r^3} \sqrt{4\cos^2\theta+\sin^2\theta} \] \[ = \frac{kP}{r^3} \sqrt{3\cos^2\theta+1} \] \[ \tan\phi = \frac{E_{\text{equatorial}}}{E_{\text{axial}}} = \frac{\tan\theta}{2} \]