Potential Energy of a Dipole in an External Field

Place a complete electric dipole -- charges $-q$ and $+q$ separated by $2l$, dipole moment $\vec{p}$ -- inside a UNIFORM external field $\vec{E}$. Because the two charges are equal and opposite, the NET force on the dipole from a uniform field is always exactly zero; but because the two equal-and-opposite forces act at two DIFFERENT points (the two ends of the dipole, separated by $2l$), they form a COUPLE, producing a net TORQUE $\vec{\tau}=\vec{p}\times\vec{E}$, of magnitude $\tau=pE\sin\theta$, that tends to rotate the dipole toward alignment with the field -- exactly the torque result already derived in Class XI.

Now suppose an external agent applies an equal-and-opposite counter-torque, so the dipole is rotated slowly (no angular acceleration, negligible kinetic energy at every instant) from some starting angle $\theta_0$ to a new angle $\theta$. The work done by this external agent, integrating the counter-torque over the angle swept, is $W_{ext}=\displaystyle\int_{\theta_0}^{\theta}pE\sin\theta'\,d\theta'=pE(\cos\theta_0-\cos\theta)$ -- and, exactly as with any conservative-force system, this work is stored as the dipole's potential energy in its new orientation: $U(\theta)-U(\theta_0)=pE(\cos\theta_0-\cos\theta)$.

Since only DIFFERENCES in PE ever matter physically, a convenient zero reference can be chosen; taking $\theta_0=90^\circ$ (dipole exactly perpendicular to the field, where $\cos\theta_0=0$) as the zero-PE reference gives the clean, standard result $U(\theta)=-pE\cos\theta=-\vec{p}\cdot\vec{E}$. …