How Is the Formula PEcos(θ) Derived for Dipoles in a Constant Electric Field?

[electrohau5]

I know that the potential energy of a dipole in a constant electric field is

P(dot)E=PEcos(θ), but I can't seem to find how they got here; its not in my textbook.

If anyone knows why please tell me.

 

[vanhees71]

A straight-forward way is to just use two charges of the same magnitude Q and opposite sign located at the origin and at \vec{r}_0 (negative charge in the origin). The total potential energy of this charge distribution in the homogeneous electric field is
V=-Q \vec{r}_0 \cdot \vec{E}.
Now the dipole moment is given for the limit \vec{r}_0 \rightarrow 0 such that Q \vec{r}_0=\vec{P}=\text{const}. This leads to
V=-\vec{P} \cdot \vec{E}.
QED.