Calculate point-charge potential and energy
Use the point-charge potential and the potential energy of a pair of point charges, preserving the signs of the source and test charges.
A Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 18. These original explanations and calculations are study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use force, charge, work, energy, potential difference, inverse-square relationships and SI units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How does point-charge potential vary with distance from the source?
What you will learn
- Use V = Q/(4πε₀r) for point-charge potential
- Relate potential energy to charge and potential
- Use Eₚ = Qq/(4πε₀r) for two point charges
Point-charge potential varies inversely with distance
For a point source charge Q in free space, the electric potential at distance r, relative to zero at infinity, is V = Q/(4πε₀r). Unlike the field-magnitude equation, the sign of Q is retained: a positive source gives positive potential and a negative source gives negative potential.
The potential is a scalar, so contributions from multiple sources add algebraically with their signs. The inverse-distance dependence differs from the inverse-square dependence of field strength.
Potential energy belongs to a pair of charges
A test charge q at potential V has electric potential energy Eₚ = qV when V is due to the other source charges. For two point charges, Eₚ = Qq/(4πε₀r). Its sign is positive for like charges and negative for unlike charges when zero energy is set at infinite separation.
Use the signed charges in the potential-energy expression. A negative value means the pair has lower energy than separated charges at infinity under this reference; it does not mean energy has a negative magnitude.
Pause and explain
A positive source charge and a negative test charge are separated by a finite distance. What sign is their potential energy relative to infinity?
Worked example
A +5.0 nC point charge is at the origin. Find V at 0.20 m and the potential energy of a −2.0 nC charge placed there.
Show the worked solution
- Use V = kQ/r with the signed source charge.
- Use Eₚ = qV or kQq/r with signed q.
- Unlike charges give negative potential energy relative to infinite separation.
Answer V ≈ +225 V; Eₚ ≈ −4.5 × 10⁻⁷ J.
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Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the potential 0.30 m from a +2.0 nC point charge. Use k = 8.99 × 10⁹ N m² C⁻².
Give me a hint
Use V = kQ/r and retain the sign of Q.
Compare my reasoning
- Q = +2.0 × 10⁻⁹ C.
- V = (8.99 × 10⁹)(2.0 × 10⁻⁹)/0.30.
- State volts and the sign.
V ≈ +60 V.
Look for these in your work
- Potential relation used
- Distance used to first power
- Sign and unit stated
A +4.0 nC charge and a −3.0 nC charge are 0.20 m apart. Calculate their electric potential energy.
Give me a hint
Use Eₚ = kQq/r with signed charges.
Compare my reasoning
- Qq = (4.0 × 10⁻⁹)(−3.0 × 10⁻⁹) = −1.2 × 10⁻¹⁷ C².
- Eₚ = (8.99 × 10⁹)(−1.2 × 10⁻¹⁷)/0.20.
- Retain the negative sign for the unlike pair.
Eₚ ≈ −5.4 × 10⁻⁷ J.
Look for these in your work
- Signed product calculated
- Potential-energy relation used
- Joule sign interpreted
A source produces potential −120 V at a point. Find the potential energy of a +5.0 μC charge placed there.
Give me a hint
Use Eₚ = qV and preserve the signs.
Compare my reasoning
- q = +5.0 × 10⁻⁶ C.
- Eₚ = qV = (5.0 × 10⁻⁶)(−120).
- State joules and interpret the sign relative to the chosen zero.
Eₚ = −6.0 × 10⁻⁴ J.
Look for these in your work
- Charge converted
- qV relation applied
- Signed energy stated
Common mistakes
- Using 1/r² for potential instead of field strength
- Dropping the source-charge sign in V
- Using only magnitudes in potential energy
- Treating negative potential energy as a negative energy magnitude rather than a reference-dependent value
What can you explain now?
How does point-charge potential vary with distance from the source?
Compare with the explanation
It is proportional to 1/r
Potential uses an inverse-distance relation.
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Make it stick
How are electric potential and potential energy related, and why do their signs matter?
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