medium difficulty9 min

Potential Energy of a System of Charges

Electrostatic potential energy of a system of charges is the work required to assemble the charges from infinity to their given positions without changing their kinetic energy.

In one line: Potential is per unit charge; potential energy is for charges together.

Concept Practice Coming Soon

Understand it

Potential energy belongs to a configuration of charges, not to a single isolated charge alone.
For two charges, U = k q1 q2/r.
For several charges, add the interaction energy of each pair once.
Like charges give positive potential energy, while unlike charges give negative potential energy when infinity is the zero reference.
For a dipole in a uniform external field, U = -pE cos θ, where θ is the angle between dipole moment and field.

Example

Two positive charges brought closer together require positive external work, so their potential energy increases. A positive and a negative charge brought closer together have negative potential energy because the electric force helps the assembly.

Formula and rule

For two charges, U = (1/(4 π ε₀)) q1q2/r, measured in joule.

For a system, U = k sum over distinct pairs(qi qj/rij).

For a dipole in uniform field, U = -pE cos θ.

Use these formulas for stationary point charges or an ideal dipole in a uniform electrostatic field.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Three charges q1 = +2 microcoulomb, q2 = +3 microcoulomb, and q3 = -1 microcoulomb are placed at the corners of an equilateral triangle of side 0.50 m.

1U = k(q1q2 + q1q3 + q2q3)/r = (9.0 × 10⁹/0.50)[6 - 2 - 3] x 10⁻¹² = 1.8 × 10⁻² J.
2The configuration has positive net energy because repulsive contribution is larger.

Method you can reuse

Draw the charge arrangement; mark all distances; write each distinct pair term once; keep charge signs; convert units to coulomb and metre; add terms; interpret the sign of total energy.

Visual tip

A configuration diagram should label q1, q2, q3 and their separations r12, r13, r23. For a dipole in field, show p, E, and angle θ.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is the potential energy of two unlike charges negative when infinity is the reference?

Think first, then show answer

It is negative because the electric attraction helps bring the charges together from infinity. The external agent would have to remove energy rather than supply positive work, so U = kq1q2/r becomes negative.

Common wrong answer

Using V = kq/r as the final answer for potential energy misses the second charge and gives volt instead of joule.

Don't mix this up

Students often count the same pair twice in a system of three or more charges. Each pair interaction must be included only once.

Exam tip

For three charges, write the three pair terms clearly: q1q2/r12, q1q3/r13, and q2q3/r23. This prevents double counting.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

It appears as two-charge or three-charge numericals, derivation of pair-energy expression, sign-based reasoning, and dipole stable or unstable equilibrium questions.

1
Enough for 1 mark
Electrostatic potential energy of a system of charges is the work required to assemble the charges from infinity to their given positions without changing their kinetic energy.
2
Add for 2 marks
Example: Two positive charges brought closer together require positive external work, so their potential energy increases. A positive and a negative charge brought closer together have negative potential energy because the electric force helps the assembly.
3
Add for 3 marks
Potential energy belongs to a configuration of charges, not to a single isolated charge alone.

Practice

Concept practice is coming soon

10 MCQs5 MinutesInstant Results

Help improve this page

Found something confusing, incorrect, or missing?