Formulas/physics/Electrostatic Potential/Potential Energy of Two-Charge System

Potential Energy of Two-Charge System

Electrostatic potential energy stored in a system of two point charges separated by distance r. This is the work done in assembling the configuration from infinity.

Class 11Class 12

Derivation

Assembly from infinity

Bring $q_1$ from infinity to its position — no work done (no field yet): $W_1 = 0$ .

Now bring $q_2$ from infinity to distance $r$ from $q_1$ . The potential at that point due to $q_1$ :

V_1 = \frac{q_1}{4\pi\varepsilon_0 r}

Work done by external agent placing $q_2$ :

W_2 = q_2 V_1 = \frac{q_1 q_2}{4\pi\varepsilon_0 r}

Total PE stored:

\boxed{U = \frac{q_1 q_2}{4\pi\varepsilon_0 r}}

Sign

$q_1 q_2 > 0$ : $U > 0$ — energy was stored pushing like charges together; released if they fly apart
$q_1 q_2 < 0$ : $U < 0$ — energy must be supplied to separate unlike charges; system is bound

Symmetry

The result is symmetric in $q_1$ and $q_2$ — it doesn't matter which charge we bring first. The PE is a property of the pair, not of either charge individually.

Note

The PE is that of the pair, not of individual charges. A single isolated charge has no PE in its own field (self-energy is infinite in classical electrostatics and is ignored at this level).

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