[NOTES/EM-03017] Potential Energy of a Dipole in External Field

 Consider a dipole located at \(r\) consisting of charges \(q\) and \(-q\) placed at points \(A_1, A_2\) having the position vectors \(\vec r_1\) and \(\vec r_2\) respectively. We use the notation \begin{equation} \vec R_i = \vec r_i -\vec r, \qquad i=1,2 \end{equation} so that \begin{equation} \vec r_i = \vec R_i + \vec r, \qquad i=1,2 \end{equation}

At this point some clarification about use of term ``dipole'' is required. By dipole we mean equal and opposite charges separated by an infinitesimal distance. An {\tt ideal dipole} will be   a limiting case when the separation goes to zero keeping the dipole moment \(\vec p\) a constant. In all applications only terms of first order in the separation of charges will be considered.

Let the dipole be placed in an external electric field \(\vec E\). Let the electric potential corresponding to the field be \(\phi(\vec r)\).  We wish to derive an expression for the potential energy of the dipole.

The electric potential \(\phi(\vec r)\) is the (mechanical) potential energy of a unit positive charge. Therefore the potential energy \(U(\vec r)\) of the dipole  will be given by \begin{equation} U(\vec r) = q \phi(r_1) - q \phi(\vec r_2). \end{equation} Let the dipole be located at \(\vec r\). We want to write the potential energy \(U\) in terms of \(\vec r\) and the dipole moment \(\vec p = q(\vec r_1-\vec r_2)\). Writing \(\vec r_{i} = \vec r + \vec R_i, i=1,2\) where \(\vec R_i = \vec r_i -\vec r\), we get


\begin{eqnarray}\label{eq04} U(\vec r) &=& q \big[ \phi(\vec r+ \vec R_1) -\phi(\vec r+ \vec R_2   )  \big] \end{eqnarray} Using the expansions \begin{equation} \phi(\vec r_i) = \phi(\vec r + \vec R_i) = \vec R_1\cdot \nabla \phi(\vec r)+ \ldots, \qquad i=1,2 \end{equation} in \eqref{eq04}, we get


\begin{eqnarray}\nonumber
 U(\vec r)
 &=& q\vec R_1\cdot \nabla \phi(\vec r) - q\vec R_2\cdot \nabla \phi(\vec r)\\\nonumber
 &=& q(\vec r_1-\vec r_2) \cdot \nabla \phi(r)\\
 &=& \vec p \cdot \nabla \phi(\vec r)\\
 &=& -\vec p \cdot \vec E(\vec r).
\end{eqnarray}


The terms second order in separation \(|\vec r_1 - \vec r_2|\) have been dropped.

The force on a dipole in an eternal electric field is given by \begin{equation} \vec F  = - \nabla U(\vec r). \end{equation} In a uniform electric field the potential energy is independent of \(\vec r\) and hence the force on the dipole will be zero.

\subsubsection{Torque on a dipole in external electric field}

The torque acting on the dipole in an external field can be computed by considering the forces on the charges \(q, -q\). Thus

\begin{eqnarray}
 \vec \tau
 &=& q \vec r_1 \times \vec E(\vec r_1) -q \vec r_2 \times \vec E(\vec r_2)
\end{eqnarray}

The above expression simplifies for uniform electric field. Using \(\vec E(\vec r_1)=\vec E(\vec r_2) =\vec r\), we get

\begin{eqnarray}
 \vec \tau
 &=& q (\vec r_1-\vec r_2) \times \vec E \\
 &=& \vec p \times \vec E.
\end{eqnarray}
To summarize : In a uniform electric field \(\vec E\) the force on a dipole is zero and the torque is \(\vec p\times \vec E\).