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पाठ्यक्रम

Electric Potential Energy of a System of Charges

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Estimated time: 10 minutes
CISCE: Class 12

Introduction

Electric potential energy (U) is the work done in assembling a system of charges by bringing them from infinity to their given positions, against or along the electric field of one another.

Think of potential energy of charges like stretching a spring. Bringing two like charges closer is like compressing a spring (energy stored, work done against repulsion) — bringing unlike charges closer is like releasing a stretched spring (energy released, work done by attraction).

CISCE: Class 12

Definition: Electric Potential Energy of a System of Charges

The total work done in bringing all the charges of the system from infinite separation to their present positions, without any change in kinetic energy.

CISCE: Class 12

Formula: Potential Energy of Two Point Charges

U = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}\]

Symbol Meaning Unit
U Potential energy of the system joule (J)
q1,q2 Point charges coulomb (C)
r Distance between charges metre (m)
ε0 Permittivity of free space 8.85 × 10−12  C2 N−1 m−2
CISCE: Class 12

Force–Energy Relation

F = −\[\frac {dU}{dr}\]​
  • Force is the negative gradient of potential energy.
  • Indicates the direction in which U decreases fastest — the natural direction of motion of a free charge.

Common Misconception: Students often think a negative sign of U means "no force" or "weak force." In reality, a negative U only signifies an attractive interaction, not the magnitude of force.

CISCE: Class 12

Stable Equilibrium Condition

A system of charges is in stable equilibrium when its potential energy is minimum.

Common Misconception: Equilibrium does NOT mean force is zero at all points — it means force is zero and any small displacement increases U (restoring tendency).

CISCE: Class 12

Potential Energy of a Three-Charge System

Setup: Three charges q1, q2, q3​ placed at the vertices of a triangle, separated by distances r12, r23, r31.

Step-by-Step Derivation

Step Action Work Done
1 Bring q1​ from infinity (no other charge present) 0
2 Bring q2​ from infinity into q1​'s field \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r_{12}}\]
3 Bring q3​ from infinity against fields of q1​ and q2​ \[\frac{1}{4\pi\varepsilon_0}\left(\frac{q_1q_3}{r_{31}}+\frac{q_2q_3}{r_{23}}\right)\]
4 Sum all pairwise terms Total U

Final Expression:

U = \[\frac{1}{4\pi\varepsilon_0}\left(\frac{q_1q_2}{r_{12}}+\frac{q_2q_3}{r_{23}}+\frac{q_3q_1}{r_{31}}\right)\]
Key Point (Board Important): The result is independent of the order in which charges are brought in, since the electrostatic force is conservative.
CISCE: Class 12

Example

Given: Two charges of 4 μC each, separated by 20 cm.

Find: Increase in potential energy.

Formula Used: U = \[\frac{1}{4\pi\varepsilon_0}\frac{q_1q_2}{r}\]

Solution: U = 9 × 109 × \[\frac{(4\times10^{-6})^2}{0.20}\]​ = 0.72 J

Answer: U = 0.72 J

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