Module 2 : Electrostatics
Lecture 9 : Potential Energy of a System of Charges
  We now bring the second charge and take it to point P$_2$. Since this charge moves in the potential field of the first charge, the work done in bringing this charge is
\begin{displaymath}W_2 = \frac{1}{4\pi\epsilon_0}q_2\frac{q_1}{r_{12}}= q_2\phi_1(P_2)\end{displaymath}
  where $\phi_1(P_2)$is the potential at P$_2$ due to the charge at P$_1$. The third charge $q_3$is to be brought to P$_3$ under the force exerted by bth $q_1$and $q_2$and is
 
\begin{displaymath}W_3 = \frac{1}{4\pi\epsilon_0}\left(\frac{q_1q_3}{r_{13}}+ \f... ..._2q_3}{r_{23}}\right) = q_3\left(\phi_1(P_3)+\phi_2(P_3)\right)\end{displaymath}
  and so on.
  The work done in assembling $N$charges $q_1, q_2,\ldots ,q_N$, located respectively at $\vec{r_1},\vec{r_2},\ldots, \vec{r_N}$is
 
\begin{eqnarray*} W &=& W_1 + W_2 + \ldots + W_N\\ &=& \frac{1}{4\pi\epsilon_0... ...1}{8\pi\epsilon_0}\sum_{i=1}^N\sum_{j=1}^N \frac{q_iq_j}{r_{ij}} \end{eqnarray*}
   
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