Physics · Ch 2 — Electrostatic Potential and Capacitance

Potential Energy of a System of Charges

2.7

Potential Energy of a System of Charges

Why Potential Energy Exists for a System of Charges

Electrostatic force is conservative. This means the work done to assemble a group of charges from infinity does not depend on the path taken. This work is stored as electrostatic potential energy of the system. The energy belongs to the configuration of charges, not to any single charge.

Step 1: Potential Energy of Two Point Charges

Consider two charges $q_1$ and $q_2$ at positions $\mathbf{r}_1$ and $\mathbf{r}_2$ , separated by distance $r_{12}$ .

Bring $q_1$ from infinity to $\mathbf{r}_1$ : No work is needed because there is no other charge to create a field.
Bring $q_2$ from infinity to $\mathbf{r}_2$ : The charge $q_1$ already creates a potential at $\mathbf{r}_2$ given by:

V_1(\mathbf{r}_2) = \frac{1}{4\pi\varepsilon_0} \frac{q_1}{r_{12}}

Work done on $q_2$ is $q_2 \times V_1(\mathbf{r}_2)$ .

This work is stored as potential energy $U$ of the two-charge system:

U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{12}}

Key points:

The result is independent of the order of bringing charges.
If $q_1 q_2 > 0$ (like charges), $U > 0$ — work is done against repulsion.
If $q_1 q_2 < 0$ (unlike charges), $U < 0$ — work is done by the attractive field (negative work by external agent).

Step 2: Generalisation to Three Point Charges

For charges $q_1, q_2, q_3$ at positions $\mathbf{r}_1, \mathbf{r}_2, \mathbf{r}_3$ :

Bring $q_1$ : Zero work.
Bring $q_2$ : Work = $\frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{12}}$ .
Bring $q_3$ : Now both $q_1$ and $q_2$ create a potential at $\mathbf{r}_3$ :

V_{1,2}(\mathbf{r}_3) = \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1}{r_{13}} + \frac{q_2}{r_{23}} \right)

Work done on $q_3$ = $q_3 \times V_{1,2}(\mathbf{r}_3)$ .

Adding all contributions gives the total potential energy:

U = \frac{1}{4\pi\varepsilon_0} \left( \frac{q_1 q_2}{r_{12}} + \frac{q_1 q_3}{r_{13}} + \frac{q_2 q_3}{r_{23}} \right)

Important: The result is path-independent — it depends only on the final positions, not on the assembly order.

Step 3: Example — Four Charges at Corners of a Square

Given: Square $ABCD$ of side $d$ , charges $+q, -q, +q, -q$ at corners $A, B, C, D$ respectively.

(a) Work to assemble the arrangement:

One valid order: bring $+q$ to $A$ , then $-q$ to $B$ , then $+q$ to $C$ , then $-q$ to $D$ .

Step (i): $W_A = 0$
Step (ii): $W_B = (-q) \times \frac{1}{4\pi\varepsilon_0} \frac{+q}{d} = -\frac{q^2}{4\pi\varepsilon_0 d}$
Step (iii): $W_C = (+q) \times \frac{1}{4\pi\varepsilon_0} \left( \frac{+q}{d} + \frac{-q}{d\sqrt{2}} \right) = \frac{q^2}{4\pi\varepsilon_0 d} \left(1 - \frac{1}{\sqrt{2}}\right)$ …

Figure 2.13Potential energy of a system of charges q1 and q2 is directly proportional to the product of charges and inversely to the distance between them.

Fig. 2.13 — Potential energy of a system of charges q1 and q2 is directly proportional to the product of charges and inversely to the distance between them.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

What the figure shows

The diagram is a minimal schematic of two point charges. A single straight line segment runs from the lower‑left to the upper‑right. At the lower‑left endpoint is a small circle labelled $q_1$ ; at the upper‑right endpoint is another small circle labelled $q_2$ . Above the middle of the segment is the label $r_{12}$ . There are no axes, no field lines, and no other markings.

Physical idea

The figure illustrates the simplest possible system of point charges: just two of them. The quantity $r_{12}$ is the distance between the two charges. The diagram is used to introduce the concept of electrostatic potential energy of a system — the work stored when the charges are brought from infinity to their present positions. The key point is that this energy depends only on the product of the charges and the separation between them, not on the path taken to assemble them.

Key formula

The textbook derives the potential energy for two point charges as:

U = \frac{1}{4\pi\varepsilon_0} \frac{q_1 q_2}{r_{12}}

where:

$U$ is the electrostatic potential energy of the system (in joules)
$\varepsilon_0$ is the permittivity of free space ( $8.85 \times 10^{-12} \, \text{C}^2/\text{N·m}^2$ )
$q_1$ and $q_2$ are the magnitudes of the two charges (with sign)
$r_{12}$ is the distance between the centres of the two charges …

Figure 2.14Potential energy of a system of three charges is given by the notation in the figure.

Fig. 2.14 — Potential energy of a system of three charges is given by the notation in the figure.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

Figure 2.14 is a schematic diagram of three point charges — $q_1$ , $q_2$ , and $q_3$ — placed at the vertices of a scalene triangle. There are no axes; the figure is a pure geometric sketch. Each charge is represented by a small circle at a vertex: $q_3$ at the top-left, $q_2$ at the right, and $q_1$ at the bottom. The sides of the triangle are labelled with the distances between the corresponding charges: $r_{12}$ (between $q_1$ and $q_2$ ) on the lower-right side, $r_{13}$ (between $q_1$ and $q_3$ ) on the lower-left side, and $r_{23}$ (between $q_2$ and $q_3$ ) on the top-left side.

The physical idea the figure teaches is that the total electrostatic potential energy of a system of three point charges is the sum of the pairwise potential energies for each distinct pair. Because electrostatic force is conservative, the work done to assemble the configuration is independent of the order in which the charges are brought from infinity. The figure visually reminds us that every pair of charges contributes a term, and the total energy depends only on the final positions (the distances $r_{12}$ , $r_{13}$ , $r_{23}$ ) and the product of the charges in each pair. …

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