[NOTES/EM-02012] Dipole in Uniform Electric Field

  Let a dipole, consisting of equal and opposite charges \(q, -q\) at positions \(\vec r_1, \vec r_2\), be placed in a uniform electric field \(\vec E\), see Fig. 1 The force on the two charges  adds up to zero.  

The torque on the dipole is given by
\begin{eqnarray}\nonumber
 \tau &=& \vec r_1 \times(q\vec E) -     
           \vec r_2 \times(q\vec E)\\\nonumber
      &=& q (\vec r_1 - \vec r_2)              
             \times\vec E\\\nonumber
      &=&  \vec p\times\vec E.
\end{eqnarray}