Take two ideal point charges of equal and opposite charge. The positive charge will be called q1, and the negative charge q2. Now separate them by a distance d. You now have yourself an electric dipole.
Now pick some point far away from the dipole. The distance from that point to charge q1 is r1, and the distance from that point to charge q2 is r2. The distance from that point to the midpoint of the two charges is r.
The potential at the point then is the combined potential produced by both charges:
kq(1/r1 - 1/r2) = kq((r2 - r1)/r2r1)
Now, as the distance r increases, d starts to become insignificant. And thus, let us assume r2 is approximately equal to r1. Thus, r2r1 can be approximated with r^2.
Well, wait. If we assume that, wouldn't r2 - r1 be equal to 0?
No, no. What nonsense. Everybody knows that r2 - r1 can be approximated with d cosθ.
θ, of course, is the angle between the line connecting the point to the midpoint of the dipole and the line connecting the two charges.
Wait what? How did you get that?
Go use the cosine law. Figure it out yourself. It's so obvious. But anyways, the potential would then be:
kqd cosθ/r^2
And just for the hell of it, let us let p be the "dipole movement". p = qd. The potential is now:
kp cosθ/r^2.
Thank you very much. And that's that. Any questions?
But wait how did you-
Nope? Okay. Moving on to capacitors...
I closed my physics textbook and put on some light-hearted Mozart. There must be easier ways of not doing my work in the yearbook.
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