# UY1: Electric Dipole

An electric dipole is a pair of charges with equal magnitude and opposite sign (a positive charge q and a negative charge -q) separated by a distance d. Consider an electric dipole placed in a uniform external electric field $\vec {E}$.

Electric dipole moment: $p = qd$.

Torque on the dipole:

$$\vec {\tau} = \vec {p} \times \vec {E}$$

$$\tau = pE \sin {\phi}$$

The torque always tends to turn $\vec {p}$ to line up with $\vec {E}$.

When a dipole changes direction in an electric field, the electric field torque does work on it.

\begin{align*} dW &= -qEd \sin{\phi} d{\phi} \\ &= -pE \sin{\phi} d{\phi} \end{align*}

In a finite displacement from $\phi _{1}$ to $\phi _{2}$, the total work done:

\begin{align*} W &= \int_{\phi _{1}}^{\phi _{2}} -pE \sin{\phi} \, \mathrm{d}\phi\\ &= pE \cos{\phi _{2}} – pE \cos{\phi _{1}}\\ &= -\left[ U(\phi_{2}) – U(\phi_{1}) \right] \end{align*}

Potential energy for a dipole in an electric field:

\begin{align*} U(\phi) &= -pE \cos {\phi}\\ &= – \vec{p} . \vec {E} \end{align*}

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