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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.

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Student-friendly explanation

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 theta, where theta is the angle between dipole moment and field.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

    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.

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

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.

Rule to remember

For two charges, U = (1/(4 pi epsilon0)) 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 theta. Use these formulas for stationary point charges or an ideal dipole in a uniform electrostatic field.

Memory hook

Potential is per unit charge; potential energy is for charges together.

Examples and method

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. U = k(q1q2 + q1q3 + q2q3)/r = (9.0 x 10^9/0.50)[6 - 2 - 3] x 10^-12 = 1.8 x 10^-2 J. The configuration has positive net energy because repulsive contribution is larger.

Method to apply

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.

Diagram support

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

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.

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

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. For two charges, U = (1/(4 pi epsilon0)) 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 theta. Use these formulas for stationary point charges or an ideal dipole in a uniform electrostatic field. 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-mark answer

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 theta, where theta is the angle between dipole moment and field. For two charges, U = (1/(4 pi epsilon0)) 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 theta. Use these formulas for stationary point charges or an ideal dipole in a uniform electrostatic field. 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. U = k(q1q2 + q1q3 + q2q3)/r = (9.0 x 10^9/0.50)[6 - 2 - 3] x 10^-12 = 1.8 x 10^-2 J. The configuration has positive net energy because repulsive contribution is larger. It appears as two-charge or three-charge numericals, derivation of pair-energy expression, sign-based reasoning, and dipole stable or unstable equilibrium questions. Using V = kq/r as the final answer for potential energy misses the second charge and gives volt instead of joule.
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