ELECTRIC POTENTIAL ENERGY OF A SYSTEM OF POINT CHARGES
Define electric potential energy of a system of charges. Derive expressions for potential energy of a system of (i) two point charges, (ii) three point charges and (iii) n point charges.
Electric potential energy of a system of point charges is defined as the total work done in bringing these point charges to their respective locations from infinity to form a system of charges.
Potential Energy of a System of Two Point Charges
Consider two point charges $q_1$ and $q_2$ initially lying at infinity. Work done to bring a charge $q$ from infinity to a point, where electric potential due to any source charge is $V(r)$ is given by
$ W = q V(r)$
When charge $q_1$ is brought from infinity to a position A, no work is done as $V(r) = 0$ at A in the absence of any source charge
When charge $q_2$ is brought from infinity to the position B, then the work done is given by
$W = q_2 \times V_1 \qquad ...(1)$
where,
$V_1 = \frac{1}{4\pi\epsilon_0} \frac{q_1}{|\vec{r}_2 - \vec{r}_1|}$
is the electric potential at position B due to charge $q_1$.
$W = q_2 \times \frac{1}{4\pi\epsilon_0} \frac{q_1}{|\vec{r}_2 - \vec{r}_1|}$
$W= \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{|\vec{r}_2 - \vec{r}_1|}$
Since $|\vec{r}_2 - \vec{r}_1| = r_{12}$, distance between charges $q_1$ and $q_2$.
$W = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$
This work done (W) is stored as the potential energy (U) of the system of two charges.
$U = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r_{12}}$
which is the expression for the electric potential energy of a system of two point charges.
From eqn. (2), we conclude that
(i) Potential energy of a system of two point charges is positive if both the charges are of same sign (either both positive or both negative i.e., like charges). Like charges repel each other and work done to bring these charges from infinity to their respective locations is positive. The variation of potential energy of a system of two point charges of same sign is shown in figure
(ii) Potential energy of a system of two point unlike charges (one positive and other negative) is negative. Unlike charges attract each other. Therefore, the work done to bring these charges from infinity to their respective locations is negative. The variation of potential energy of a system of two point charges of opposite signs is shown in figure
Potential Energy of a System of Three Point Charges
Let us consider a system of three point charges $q_1$, $q_2$ and $q_3$ such that $q_1$, $q_2$ and $q_3$ are initially at infinity.
Let $\vec{r}_1$, $\vec{r}_2$ and $\vec{r}_3$ are position vectors of charges $q_1$, $q_2$ and $q_3$ respectively.
STEP 1. No work is done to bring charge $q_1$ from infinity to the position A, when charges $q_2$ and $q_3$ are at infinity and there is no other charge near the charge $q_1$.
STEP 2. Work done to bring charge $q_2$ from infinity to position B, when charge $q_1$ is at position A is given by
$W_{12} = q_2 \times V_1$
where $V_1 = \frac{1}{4\pi\epsilon_0} \frac{q_1}{|\vec{r}_2 - \vec{r}_1|} = \frac{1}{4\pi\epsilon_0} \frac{q_1}{r_{12}}$
It is the electric potential at the position B due to charge $q_1$.
$W_{12} = q_2 \times \frac{1}{4\pi\epsilon_0} \frac{q_1}{r_{12}} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$...(i)
STEP 3. Work done in bringing charge $q_3$ (in the presence of charge $q_1$ and $q_2$ at their respective positions) from infinity to point C is given by,
$W_{123} = V_1 \times q_3 + V_2 \times q_3$...(ii)
where,
$V_1$Electric potential at C due to charge $q_1 = \frac{1}{4\pi\epsilon_0} \frac{q_1}{|\vec{r}_3 - \vec{r}_1|} = \frac{1}{4\pi\epsilon_0} \frac{q_1}{r_{13}}$
and
$V_2$ = Electric potential at C due to charge $q_2 = \frac{1}{4\pi\epsilon_0} \frac{q_2}{|\vec{r}_3 - \vec{r}_2|} = \frac{1}{4\pi\epsilon_0} \frac{q_2}{r_{23}}$
Hence eqn. (ii) becomes
$W_{123} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_3}{r_{13}} + \frac{1}{4\pi\epsilon_0} \frac{q_2 q_3}{r_{23}}...(iii)$
STEP 4. The total work done to bring all the three charges from infinity to their respective positions to form the system of charges is given by
$W = W_{12} + W_{123}$
or
$W = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}} + \frac{1}{4\pi\epsilon_0} \frac{q_1 q_3}{r_{13}} + \frac{1}{4\pi\epsilon_0} \frac{q_2 q_3}{r_{23}}$
$= \frac{1}{4\pi\epsilon_0} \left[ \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right]$
But work done, $W =$ Potential energy $(U)$ of the system of three charges
$U = \frac{1}{4\pi\epsilon_0} \left[ \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right]$...(iv)
or
$U = \frac{1}{2} \left[ \frac{1}{4\pi\epsilon_0} \sum_{i=1}^3 \sum_{\substack{j=1 \\ j \neq i}}^3 \frac{q_i q_j}{r_{ij}} \right]$...(v)
The factor $(1/2)$ is introduced in eqn. (v) because each term gets counted twice in the above manner of writing the expression. For example, when $i=1, j=2$ and $i=2, j=1$, contribution is got from same pair of charges, $q_1$ and $q_2$. As such factor $1/2$ is required to include only one term in each pair.
Rewriting eqn. (v), we get
$U = \frac{1}{2} \sum_{i=1}^3 q_i \left( \sum_{\substack{j=1 \\ j \neq i}}^3 \frac{1}{4\pi\epsilon_0} \frac{q_j}{r_{ij}} \right)$
or
$U = \frac{1}{2} \sum_{i=1}^3 q_i V_i$ ...(vi)
where, $V_i$ is potential at $\vec{r}_i$ due to all other charges.