Electric Potential Energy: Study Guide (Physics 2 10.4)

Energy stored in a system of charges

In Physics 1 you stored energy by lifting a book: you did work against gravity, and the Earth–book system gained gravitational potential energy. Charges work the same way. Push two positive charges together and you do work against their repulsion. That energy is stored in the system of the two charges as electric potential energy, U_E.

The definition uses a fixed starting point:

The electric potential energy of two point charges equals the work an external force must do to bring them from infinitely far apart to their present positions.

Two details matter.

• The charges are moved slowly, so they start and end at rest. Then all of the external work goes into U_E, not into kinetic energy.
• At infinite separation the charges do not interact, so U_E = 0 there. This is the zero the course uses.

U_E belongs to the pair, not to either charge on its own. It is a scalar: it has a sign but no direction.

The equation and its sign

For two point charges q₁ and q₂ a distance r apart:

U_E = k q₁ q₂ / r = (1/(4πε₀)) q₁ q₂ / r

with k = 9.0 × 10⁹ N·m²/C². Put the signs of the charges into the equation. The sign of the answer tells you about the system.

• Like charges (both + or both −): q₁q₂ > 0, so U_E > 0. They repel, so the external force had to push them together. Release them and they fly apart, turning U_E into kinetic energy.
• Unlike charges: q₁q₂ < 0, so U_E < 0. They attract, so the external force had to hold them back as they came together: it did negative work. To separate them again you must add |U_E|.

A negative U_E does not mean “no energy”. It means the system has less energy than when the charges were far apart.

Comparing with gravitational potential energy

In Physics 1 the gravitational potential energy of two masses is U_G = −Gm₁m₂/r, with the same zero at infinite separation. It has the same 1/r form as U_E. The difference is the sign. Gravity always attracts, so U_G is always negative. Charges can attract or repel, so U_E can be negative or positive, and the signs of the charges decide which.

Graphs of U_E against separation

Figure 1 shows U_E against r for two fixed charges. The pattern comes straight from U_E ∝ 1/r.

• Both curves approach zero at large r, because U_E = 0 at infinite separation.
• Both get very steep at small r. U_E grows without limit as the charges get very close.
• The two curves are mirror images in the r-axis: swapping the sign of one charge flips the sign of U_E.
• Moving to the right on the solid curve, U_E falls: the like charges lose energy as they separate. Moving to the right on the dashed curve, U_E rises toward zero: unlike charges gain energy as you pull them apart.

Background: the steeper the graph at a point, the larger the force between the charges there. This is why both curves are steep at small r, where the Coulomb force is large.

Predicting changes

Because U_E = kq₁q₂/r, you can predict factors of change without numbers.

Change	Effect on U_E	Compare: effect on the force
Separation doubled	× 1/2	× 1/4
Separation tripled	× 1/3	× 1/9
Separation halved	× 2	× 4
One charge doubled	× 2	× 2
Both charges doubled	× 4	× 4

The most common error is to use 1/r² for energy. Force goes as 1/r²; energy goes as 1/r.

Systems of more than two charges

The total electric potential energy of a system is the sum of U_E for every pair of charges. The course asks for systems of up to four point charges.

Why pairs? Build the system one charge at a time. Bringing in the first charge costs nothing, because there is nothing to push against. The second costs kq₁q₂/r₁₂. The third has to be pushed against both of the first two, so it costs kq₁q₃/r₁₃ + kq₂q₃/r₂₃. Every pair appears exactly once.

Number of charges	Number of pairs
2	1
3	3
4	6

Each pair keeps its own sign, so some terms are positive and some negative. Do not count a pair twice, and do not forget pairs that are not next to each other (such as the diagonals of a square).

Work done to rearrange charges

If the charges start and end at rest, the work done by an external force equals the change in the system’s electric potential energy:

W_ext = ΔU_E = U_E,final − U_E,initial

• W_ext > 0: you must push or pull, adding energy to the system (for example, separating unlike charges).
• W_ext < 0: the system does work on whatever holds the charges back (for example, letting like charges move apart slowly).

If nothing holds the charges back, the change in U_E becomes kinetic energy instead. Topic 10.7 develops this.

Worked example 1: separating two charges

Question. A +3.0 μC charge and a −2.0 μC charge are held 0.15 m apart. (a) Find the electric potential energy of the system. (b) How much work must an external force do to move them slowly to 0.45 m apart? (c) How much work would separate them completely?

1. (a) U_E = (9.0 × 10⁹)(3.0 × 10⁻⁶)(−2.0 × 10⁻⁶) ÷ 0.15 = −0.36 J.
2. (b) At 0.45 m the separation is 3 times larger, so U_E is 3 times smaller: U_E = −0.36 ÷ 3 = −0.12 J. (Direct check: (9.0 × 10⁹)(3.0 × 10⁻⁶)(−2.0 × 10⁻⁶) ÷ 0.45 = −0.12 J.)
3. W_ext = ΔU_E = −0.12 J − (−0.36 J) = +0.24 J.
4. (c) At infinite separation U_E = 0, so W_ext = 0 − (−0.36 J) = +0.36 J.

Interpretation. Both works are positive, as they must be: unlike charges attract, so you have to pull to separate them. The system’s energy rises from −0.36 J toward zero. Notice that the first 0.30 m of pulling needs two-thirds of the total energy. The force is strongest when the charges are close.

Worked example 2: three charges

Question. Three point charges sit at the corners of the right triangle in Figure 2: q₁ = +2.0 μC at A, q₂ = +3.0 μC at B, 0.30 m from A, and q₃ = −5.0 μC at C, 0.40 m from A. (a) Find the total electric potential energy. (b) How much external work is needed to remove q₃ slowly to a very large distance?

1. The distance BC is the hypotenuse: √(0.30² + 0.40²) = 0.50 m.
2. Pair 1–2: U₁₂ = (9.0 × 10⁹)(2.0 × 10⁻⁶)(3.0 × 10⁻⁶) ÷ 0.30 = +0.18 J.
3. Pair 1–3: U₁₃ = (9.0 × 10⁹)(2.0 × 10⁻⁶)(−5.0 × 10⁻⁶) ÷ 0.40 = −0.225 J.
4. Pair 2–3: U₂₃ = (9.0 × 10⁹)(3.0 × 10⁻⁶)(−5.0 × 10⁻⁶) ÷ 0.50 = −0.27 J.
5. (a) Total: U_E = 0.18 − 0.225 − 0.27 = −0.315 J.
6. (b) With q₃ gone, only the pair 1–2 is left, so U_E = +0.18 J. W_ext = 0.18 − (−0.315) = +0.495 J.

Check. The work in (b) equals −(U₁₃ + U₂₃) = −(−0.495 J), the two pairs that disappear. It is positive, because q₃ is attracted to both positive charges and has to be pulled away.

Worked example 3: predicting a change

Question. Two charged spheres, far smaller than their separation, have U_E = +0.80 J. One sphere’s charge is tripled and the separation is doubled. What is the new U_E?

1. U_E ∝ q₁q₂/r. Tripling one charge multiplies U_E by 3. Doubling r multiplies it by 1/2.
2. New U_E = 0.80 J × 3 × 1/2 = +1.2 J.

Check. The sign stays positive because neither charge changed sign. Using 1/r² by mistake would give 0.60 J.

Common misconceptions

• “The charge has potential energy.” U_E belongs to the system of interacting charges. A single charge on its own in empty space has no electric potential energy.
• Dropping the signs. Using |q₁| and |q₂| makes every U_E positive. An attracting pair must have U_E < 0.
• Using 1/r² instead of 1/r. That is the force law, not the energy.
• “Negative energy is impossible.” It just means less energy than at infinite separation. The zero is a choice.
• Counting pairs wrongly. Four charges have six pairs, including the diagonals of a square. Counting each pair from both ends doubles the answer.
• “U_E = 0 means no forces act.” Total U_E can be zero while forces act. For example, two +2.0 μC charges and one −1.0 μC charge at the corners of an equilateral triangle of side 0.20 m have U_E = 0.18 − 0.09 − 0.09 = 0 J, yet every charge feels a force.
• Mixing up the external work and the work done by the electric force. When the charges start and end at rest, the external work is +ΔU_E and the work done by the electric force is −ΔU_E. Say which one you mean.
• Mixing up U_E and potential. U_E is in joules and needs at least two charges. Potential (Topic 10.5) is energy per unit charge, in volts.

Where this leads

Topic 10.5 divides U_E by the charge to get the electric potential, a property of a point in space: see Electric Potential. Topic 10.7 uses changes in U_E to find the speed of moving charges. Test yourself now with the practice questions, then use the revision notes and the topic checklist. For the field picture behind these forces, revisit Electric Fields.