Electric field generated from dipoles

[Repetit]

I'm having a little difficulty understanding the electric field generated from electric dipoles. As far as I know electric fields are generated on positive charges and terminated on negative charges. Since a dipole is a positive charge and a negative charge separated by a very small distance, why doenst they just simply cancel each other out?

Thanks
René

 

[Tide]

If the positive and negative charge were point charges at exactly the same location then they would cancel out. However, if the separation between them is finite (but as small as you like) then the cancellation is not exact.

Mathematically, it's like this:

$\frac {1}{x}$ and $- \frac {1}{x}$ cancel exactly when added together (we ignore x = 0 for this discussion) but $\frac {1}{x-\epsilon}$ and $-\frac {1}{x+\epsilon}$ do not cancel.