Electric field of a dipole, two different equations?

[gareth182]

Electric field of a dipole, two different equations??

Hi,

I've been taught in uni how to derive the electric field of a dipole to be

E= k p [2cos theta r(hat) + sin theta theta(hat)]/(r^3)

where k=1/(4pi e0), p=qd= dipole moment, (hat) terms are unit vectors and r is the distance between the halfway point of the dipole and the point of observation.

However, on most internet pages the electric field is expressed as

E= k[3(p.r(hat))r(hat) - p]/(r^3)

can anyone help me convert the first electric field equation to the second?

I'm guessing it may involve using r(hat)=r/r to convert the trigonometric terms back to dot products...

 

[Bill_K]

Sorry, I've got to use underlines instead of hats to denote unit vectors. The two expressions you have are

(1) E = kp[2 cos θ r + sin θ θ]/r³ and
(2) E = kp[3 (z·r) r - z]/r³ = kp[3 cos θ r - z]/r³

Geometry tells us that z = cos θ r - sin θ θ, so plug that in (2) and you get (1).