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.