Electric Potential Energy

Electric potential energy is the energy stored in a system of charges because of their relative positions. It depends on both the positions of the charges and their magnitudes — the same way gravitational potential energy depends on the relative positions of masses. Just as lifting a mass higher increases its gravitational potential energy, changing the separation between two charges changes the electric potential energy stored between them.

Key Takeaways

• Electric potential energy: U = kQq/r, the energy stored due to the relative positions of two charges — analogous to gravitational potential energy.

• U can also be defined as the work required to bring a charge from infinitely far away to a given point (same unit: the joule).

• Like charges → positive U (repulsive, energy must be added); unlike charges → negative U (attractive, energy is released).

• U depends on a pair of charges, distinct from electric potential (V), which describes a single point in space.

The Electric Potential Energy Formula

For two charges, electric potential energy is given by:

U = kQq/r

where:

• U is the electric potential energy,

• k is Coulomb's constant,

• Q and q are the magnitudes of the two charges,

• r is the distance between them.

U is proportional to the product of the two charges, and inversely proportional to the distance between them. As the charges move closer together, the magnitude of the potential energy increases significantly.

Electric Potential Energy and Work

Work and energy share the same unit — the joule — so electric potential energy for a charge at a given point in an electric field can also be defined as the amount of work required to bring that charge from infinitely far away to that point.

This follows directly from Coulomb's Law and the work formula. Since Fe = kqQ/r² and W = Fd·cos(θ), if the displacement d is taken as the separation r between the charges, and the force and displacement vectors are parallel, the result is exactly the electric potential energy relationship above.

The Sign of Electric Potential Energy

The sign of U reveals the nature of the interaction between the two charges:

• Like charges (both positive, or both negative): U is positive. The charges repel each other, so energy must be put in to bring them closer together — potential energy increases.

• Unlike charges (one positive, one negative): U is negative. The charges attract each other, so energy is released as they move closer together — potential energy decreases.

MCAT Callout — Worked Example: Electric Potential Energy of Two Charges: Two charges of +3×10⁻⁶ C and +2×10⁻⁶ C, separated by 0.3 m:

U = (8.99×10⁹)(3×10⁻⁶)(2×10⁻⁶)/0.3 ≈ 0.180 J

Positive, as expected for two like (positive) charges.

Now replace the second charge with −2×10⁻⁶ C, same distance:

U = (8.99×10⁹)(3×10⁻⁶)(−2×10⁻⁶)/0.3 ≈ −0.180 J

Negative, as expected for an attractive, unlike-charge pair.

MCAT Callout — Potential Energy vs. Potential: Electric potential energy (U) describes a pair of charges — it depends on both charges involved. Electric potential (V, covered next) describes a single location in space, independent of any particular test charge placed there.

Common MCAT Mistakes

• Confusing electric potential energy with electric potential. U (potential energy, joules) depends on a pair of charges; V (potential, volts) describes a single point in space independent of any test charge placed there. They're related but not interchangeable.

• Ignoring the sign of the charges when computing U. The sign of U isn't a magnitude detail — it tells you whether the interaction is repulsive (positive U, like charges) or attractive (negative U, unlike charges).

• Treating U like force and assuming it always decreases as charges approach. For like charges, U increases (becomes more positive) as separation decreases, since work must be done against the repulsion — the opposite of the unlike-charge case.

• Forgetting the inverse relationship with distance. U = kQq/r scales with 1/r, not 1/r² like the Coulomb force — halving the distance doubles the magnitude of U, it doesn't quadruple it.

MCAT-Style Concept Check

Question: A +4×10⁻⁶ C charge and a −6×10⁻⁶ C charge are separated by 0.2 m. What is the electric potential energy of this system, and what does its sign indicate?

• A) +1.08 J; the charges repel each other

• B) −1.08 J; the charges attract each other

• C) +0.54 J; the charges repel each other

• D) −0.54 J; the charges attract each other

Answer: B

Explanation: U = kQq/r = (8.99×10⁹)(4×10⁻⁶)(−6×10⁻⁶)/(0.2) ≈ −1.08 J. The negative sign confirms the charges are unlike (one positive, one negative) and therefore attract — energy is released as they move closer together.

FAQ

What is electric potential energy?

Electric potential energy (U) is the energy stored in a system of two charges due to their relative positions, given by U = kQq/r. It's analogous to gravitational potential energy, which depends on the relative positions of masses.

How is electric potential energy related to work?

Electric potential energy at a point can be defined as the work required to bring a charge from infinitely far away to that point. This follows directly from combining Coulomb's Law (Fe = kqQ/r²) with the work formula (W = Fd·cos(θ)).

Why is electric potential energy positive for like charges and negative for unlike charges?

Like charges repel, so energy must be put into the system to force them closer together, making U positive. Unlike charges attract, so energy is released as they move closer together, making U negative.

What's the difference between electric potential energy and electric potential?

Electric potential energy (U) depends on a pair of charges and their separation. Electric potential (V) describes a single point in space and doesn't depend on any particular test charge being present there — it's the potential energy per unit charge at that point.