Difficulty: Intermediate | Prerequisites: Coulomb's law, electric field basics (Chapters 21-22 in most texts)
Electric potential energy is the energy stored in a system of charges because of their positions relative to one another. It is the electrostatic equivalent of gravitational potential energy: just as lifting a mass against gravity stores energy, moving a charge against an electric field stores energy. This topic builds directly on Coulomb's law and sets the stage for electric potential (voltage), which comes next. If you are comfortable calculating forces between point charges, you have what you need here.
Electric potential energy between two point charges depends on the product of the charges divided by their separation distance, scaled by Coulomb's constant. For multi-charge systems, you sum the PE of every unique pair. Changes in PE tell you how much work the electric field does (or you must do) when charges move.
Electric potential energy (U or PE)
The energy a system of charges possesses due to their relative positions. Measured in joules (J).
In simple terms, this is the stored energy that would be released (or absorbed) if you let the charges move freely.
Coulomb's constant (k)
The proportionality constant in Coulomb's law, equal to 8.99 x 10⁹ N·m²/C². Also written as 1/(4πε₀).
Think of it as the conversion factor that turns charge and distance into force or energy.
Permittivity of free space (ε₀)
A fundamental constant, 8.85 x 10⁻¹² C²/(N·m²), that describes how easily electric fields permeate a vacuum.
In simple terms, it sets the baseline for how strongly charges interact in empty space.
Change in potential energy (ΔPE or ΔU)
The difference in electric potential energy between two configurations. ΔU = U_final - U_initial.
Think of it as: how much energy was gained or lost by the charge system when something moved from point A to point B.
Reference point (U = 0 at infinity)
By convention, the potential energy of a pair of point charges is defined as zero when they are infinitely far apart. All PE values are measured relative to this baseline.
In simple terms, "zero energy" means the charges are so far apart they do not interact at all.
The electric potential energy of a pair of point charges q₁ and q₂ separated by distance r is:
U = k q₁ q₂ / r
The sign matters. Like charges (both positive or both negative) yield positive U, meaning you had to do work to push them together. Opposite charges yield negative U, meaning the system released energy as they came together.
U depends only on the scalar distance r between charges, not direction. There is no vector component to PE.
When a charge moves from position A (distance r_A from another charge) to position B (distance r_B):
ΔU = k q₁ q₂ (1/r_B - 1/r_A)
If ΔU is negative, the electric field did positive work on the charge (it moved "downhill" in energy). If ΔU is positive, an external agent had to do work to move it.
From the homework: with q₁ = 0.2 μC and q₂ = -2.8 μC, moving from r_A = 5.1 cm to r_B = 2 cm, you plug into the formula above. Keep the signs on the charges and convert cm to metres.
For three or more charges, the total PE is the sum of the PE of every unique pair:
U_total = U₁₂ + U₁₃ + U₂₃ + ...
With N charges, the number of unique pairs is N(N-1)/2. For three charges, that is three pairs.
From the homework (Problem 5): three charges of -2.8 μC each, placed at specific positions. You calculate U for each pair using their separation distances (2 cm, 5.1 cm, 7.1 cm), then add.
A common setup in this course places charges along a line or at vertices of a triangle. The geometry determines r for each pair.
The work done by the electric force equals the negative of the change in PE:
W_elec = -ΔU
If you are asked for the work done by an external agent (you pushing the charge), it is the opposite:
W_external = +ΔU
This sign convention trips people up. The field does positive work when PE decreases. You do positive work when PE increases.
PE between two point charges:
U = k q₁ q₂ / r
where k = 8.99 x 10⁹ N·m²/C², and r is the centre-to-centre distance in metres.
Change in PE when a charge moves:
ΔU = k q₁ q₂ (1/r_B - 1/r_A)
where r_A is the initial separation and r_B is the final separation.
Total PE for a system of N point charges:
U_total = Σ (over all unique pairs i < j) k qᵢ qⱼ / rᵢⱼ
For 3 charges: U_total = k(q₁q₂/r₁₂ + q₁q₃/r₁₃ + q₂q₃/r₂₃)
Work-energy relation:
W_field = -ΔU
W_external = +ΔU
Electric potential energy governs how capacitors store energy, which is the basis of camera flashes, defibrillators, and power conditioning in electronics. The same principle underlies the energy released when ionic bonds form in chemistry.
Students often treat PE as always positive. It is not. Opposite charges have negative PE, and that sign carries physical meaning (you would need to add energy to pull them apart).
Confusing force with energy. Coulomb's law gives force (a vector, with direction). The PE formula gives a scalar (no direction). They share the same ingredients but are different quantities.
Forgetting to convert centimetres to metres. The homework uses cm throughout, but k is in SI units. Missing this conversion is the single most common arithmetic error on exams.
Mixing up which distance is r_A and which is r_B in the ΔU formula. r_A is where the charge starts, r_B is where it ends. Swapping them flips the sign of your answer.
⚠️ Sign errors on PE are the number-one mark killer. Always carry the signs of q₁ and q₂ through the calculation, do not take absolute values.
⚠️ Multi-charge PE problems (three or more charges) are a staple of exams. You will be expected to identify all unique pairs and sum correctly.
⚠️ The relationship W = -ΔU is frequently tested as a conceptual question. Know which sign corresponds to "the field did the work" versus "an external agent did the work."
True or False: The electric potential energy of two positive charges is negative.
False. Two like charges have positive PE.
Fill in the blank: The PE of two charges is zero when they are separated by ______.
Infinity (r → ∞)
True or False: If ΔU is negative when a charge moves, the electric field did positive work on it.
True. W_field = -ΔU, so negative ΔU means positive work by the field.
How many unique pairs exist in a system of 4 point charges?
6 pairs. N(N-1)/2 = 4(3)/2 = 6.
Q: Two charges, q₁ = +3 μC and q₂ = -5 μC, are initially 10 cm apart. They are moved to 4 cm apart. Is ΔU positive or negative, and did the electric field do positive or negative work?
A: Since the charges are opposite, U is negative and becomes more negative as they get closer (1/r increases, and the product q₁q₂ is negative). So ΔU is negative. The field did positive work (W = -ΔU > 0), meaning the charges were attracted together naturally.
Q: Three identical charges of +2 μC sit at the corners of an equilateral triangle with side length 5 cm. What is the total PE of the system?
A: Three unique pairs, each with the same charges and same distance. U_total = 3 x k(2x10⁻⁶)²/(0.05) = 3 x (8.99x10⁹)(4x10⁻¹²)/(0.05) = 3 x 0.719 = 2.16 J.
Q: A charge q₂ = -2.8 μC is moved from 5.1 cm to 2.0 cm away from a charge q₁ = +0.2 μC. Calculate ΔU.
A: ΔU = k q₁ q₂ (1/r_B - 1/r_A) = (8.99x10⁹)(0.2x10⁻⁶)(-2.8x10⁻⁶)(1/0.02 - 1/0.051). Compute the bracket: 50 - 19.6 = 30.4. Then ΔU = (8.99x10⁹)(-5.6x10⁻¹³)(30.4) ≈ -0.153 J. The negative sign confirms the field pulled the opposite charges together.
This connects directly to electric potential (voltage), which is PE per unit charge: V = U/q. Once you are comfortable with PE between point charges, voltage is a short step. It also feeds into capacitor energy storage (U = ½CV²), which you will meet in a few weeks.
The work-energy ideas here mirror the conservative force framework from mechanics. If you understood gravitational PE and conservation of energy, the same logic applies with Coulomb's force replacing gravity.
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