Source: Check 3, University Physics: Electricity & Magnetism
Tags: electric potential energy, Coulomb potential energy, point charge energy, kQq/r, work-energy theorem, superposition of potential energy, PHYS 212, electrostatics
Difficulty: Intermediate | Prerequisites: Coulomb's law, electric fields, work and energy from mechanics
Electric potential energy is the energy stored in the configuration of charges. It builds directly on Coulomb's law (which gave you force) by asking: how much work does it take to assemble this arrangement of charges? This concept sits between electric force (already covered) and electric potential (coming next), and it is the bridge that lets you use conservation of energy instead of Newton's second law to solve electrostatics problems. You should already be comfortable with Coulomb's law, the superposition principle, and the work-energy theorem from mechanics.
The electric potential energy between two point charges is U = kQq/r, and it depends only on separation, not on the path taken. When you move a charge, the change in potential energy is ΔU = kQq(1/r_final − 1/r_initial). Systems of more than two charges use pairwise superposition, and a free charge always moves in the direction that lowers its potential energy.
Electric potential energy (U)
The energy associated with the configuration of a system of charges. For two point charges, U = kQq/r. Think of it as the "stored work" required to bring charges from infinitely far apart to their current positions.
Coulomb constant (k)
The proportionality constant in Coulomb's law, k = 8.99 × 10⁹ N·m²/C². It sets the scale for how strong electrostatic interactions are.
Change in potential energy (ΔU)
The difference in potential energy between a final and initial configuration: ΔU = U_final − U_initial. In simple terms, this tells you how much energy was added to or released from the system when charges were rearranged.
Superposition of potential energy
The total potential energy of a multi-charge system is the sum of the potential energies of every unique pair. Think of it as adding up the energy "cost" of assembling the system one charge at a time.
Path independence
The work done by the electric force depends only on start and end positions, not the route taken. This is because the electrostatic force is conservative, just like gravity.
Zero of potential energy
By convention, U = 0 when charges are infinitely far apart. This is the reference point: all potential energy values are measured relative to this separation.
Electric potential (V)
The potential energy per unit charge at a point in space: V = U/q = kQ/r. You will meet this properly in the next topic, but it appears here because it determines what happens when you add a third charge to an existing system.
For two point charges Q and q separated by distance r:
U = kQq / r
U is positive when both charges have the same sign (repulsive configuration, energy was required to push them together).
U is negative when the charges have opposite signs (attractive configuration, energy was released as they came together).
The formula uses the scalar distance r, not a vector. No direction is involved.
When charge +q moves from distance r₁ to distance r₂ from a fixed charge +Q:
ΔU = kQq (1/r₂ − 1/r₁)
If the charge moves farther away (r₂ > r₁) from a like charge, ΔU is negative: the system loses potential energy.
If the charge moves closer to a like charge (r₂ < r₁), ΔU is positive: energy is stored.
The displacement R (the straight-line distance the charge travels) does not appear in the formula. Only the initial and final separations from the source charge matter.
For three charges q₁, q₂, q₃, the total potential energy is the sum of all pairwise terms:
U_total = kq₁q₂/r₁₂ + kq₁q₃/r₁₃ + kq₂q₃/r₂₃
Each pair is counted once.
Adding a third charge to an existing two-charge system changes the total energy by the sum of the new charge's interactions with each existing charge.
Two charges of equal magnitude but opposite sign (a dipole) are placed symmetrically about a point A, both at the same distance d from A.
The electric potential at A due to this dipole is zero, because the positive contribution (+kq/d) and negative contribution (−kq/d) cancel exactly.
Bringing any third charge to point A from infinity requires zero work, because V = 0 there. The change in potential energy of the system from the new charge's interactions is zero regardless of the sign or magnitude of the third charge.
However, the original pair's potential energy (kq₁q₂/r₁₂) is unchanged and still contributes to the total.
The answer to "does the total energy change?" depends on the sign of the third charge only insofar as it interacts with the existing pair, and at the midpoint of a dipole those interactions cancel. So the potential energy of the collection does not change.
Consider two charges: +q (small, positive) and −2q (negative, twice the magnitude), separated by distance d.
The existing system has potential energy U₁₂ = k(+q)(−2q)/d, which is negative.
To add a third charge q₃ from infinity without changing the total energy, the work done on q₃ must be zero. This means q₃ must be placed at a location where the net electric potential from the existing charges is zero: V₁ + V₂ = 0 at that point.
For charges of unequal magnitude, V = 0 along a specific surface (not just the midpoint). You can find it by solving kq/r₁ + k(−2q)/r₂ = 0, which gives r₂ = 2r₁.
Because V = 0 locations exist regardless of the sign of q₃, the answer is: yes, you can do this no matter what the sign of the third charge is (option c).
A point charge released from rest in an electric field will always begin to move in the direction that decreases its potential energy.
This follows from conservation of energy: kinetic energy increases, so potential energy must decrease.
A positive charge moves in the direction of the electric field (from high potential to low potential).
A negative charge moves opposite to the electric field (from low potential to high potential).
In both cases, the charge moves toward lower potential energy.
Quantity | Formula | Notes |
|---|---|---|
PE of two point charges | U = kQq / r | r is the separation; sign of U follows from signs of Q and q |
Change in PE | ΔU = kQq(1/r₂ − 1/r₁) | r₁ = initial separation, r₂ = final separation |
Total PE (three charges) | U = kq₁q₂/r₁₂ + kq₁q₃/r₁₃ + kq₂q₃/r₂₃ | Sum over all unique pairs |
Electric potential (single charge) | V = kQ / r | Scalar, used to find PE when a new charge is placed at that point: U = qV |
Coulomb constant | k = 8.99 × 10⁹ N·m²/C² | Also written as 1/(4πε₀) |
The concept of electric potential energy is central to how capacitors store energy in circuits, from the tiny capacitors in your phone's processor to the large ones in defibrillators. It also underpins the energy calculations in molecular chemistry: the stability of ionic bonds (like NaCl) comes down to the negative potential energy between oppositely charged ions.
Students often think the path taken matters when calculating ΔU. It does not. The electrostatic force is conservative, so only the starting and ending positions determine the energy change.
Students sometimes confuse potential energy (U, measured in joules, a property of the system) with electric potential (V, measured in volts, a property of the location in space). U = qV ties them together, but they are different quantities.
A common error is forgetting that U can be negative. Unlike kinetic energy, potential energy has no requirement to be positive. Opposite charges have negative PE, meaning work must be done to pull them apart.
Students frequently assume that the displacement distance R (how far a charge physically travels) appears in the ΔU formula. It does not. Only the distances from the source charge matter.
⚠️ The formula ΔU = kQq(1/r₂ − 1/r₁) is heavily tested. Be careful with the order of r₂ and r₁: r₂ is the final separation, r₁ is the initial separation.
⚠️ Questions about adding a charge at a point where V = 0 are a classic conceptual trap. At such a point, the work done to bring any charge from infinity is zero, regardless of the sign of the new charge.
⚠️ "Which direction does a released charge move?" always has the same answer: toward lower potential energy. Do not confuse this with "toward lower electric potential" (which is only true for positive charges).
⚠️ Multi-charge energy problems require you to list all unique pairs. For three charges, that is three pairs. For four charges, six pairs. Missing a pair is the most common arithmetic error.
True or false: The electric potential energy between two like charges is always positive.
True. Like charges repel, and energy must be added to bring them together from infinity.
Fill in the blank: A charge released from rest in an electric field moves in the direction that ________ its potential energy.
Decreases.
True or false: If the electric potential at a point is zero, no charge can be placed there.
False. A charge can be placed there, and the work to bring it from infinity is zero.
Fill in the blank: The total potential energy of a system of three charges involves ________ pairwise terms.
Three.
Q: A charge +q is moved from distance r₁ to distance r₂ from a fixed charge +Q. What is the change in electric potential energy?
A: ΔU = kQq(1/r₂ − 1/r₁). If r₂ > r₁ (charge moves away), ΔU < 0 and the system loses energy.
Q: Two equal-and-opposite charges are placed at equal distances from point A. A third charge is brought to point A from infinity. How does the system's total potential energy change?
A: It does not change. The potential at A is zero (contributions from the two charges cancel), so bringing any charge there from infinity requires zero work.
Q: Two point charges, +q and −2q, are separated by a distance d. Is it possible to find a location where a third charge can be placed without changing the total potential energy, and does the sign of the third charge matter?
A: Yes, such a location exists (where V = 0 due to the existing charges), and the sign of the third charge does not matter. Any charge placed at a point of zero potential adds zero energy to the system.
Q: A positive charge is released from rest in a uniform electric field. In which direction does it move, and why?
A: It moves in the direction of the electric field, which is the direction of decreasing potential energy. Conservation of energy requires that kinetic energy increases at the expense of potential energy.
Q: A negative charge is released from rest in a uniform electric field. Does it move toward higher or lower electric potential?
A: It moves toward higher electric potential (opposite to the field direction), but crucially, it still moves toward lower potential energy. U = qV, and with q negative, lower U corresponds to higher V.
This material connects directly to electric potential (V), which you will study next. Potential energy is the "system" version; electric potential is the "per unit charge" version, and nearly every potential problem reduces to a potential energy problem once you multiply by q. It also links back to Gauss's law: knowing the field lets you compute the potential, which lets you compute the energy. Later in the course, capacitance and energy storage in capacitors are direct applications of the potential energy ideas here.
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