Coulomb's Law and Electric Field

Coulomb's Law, Electric Field and Electric Potential: What Changes With Distance

By StudyStore · Published 2026-11-05 · 2 min read · 498 words

Four quantities describe point charges, and the MCAT tests which one scales with which power of distance. Force and field go as 1/r². Potential and potential energy go as 1/r. Force and field are vectors, potential and energy are scalars.

The four formulas

Quantity	Formula (point charge)	Scaling	Type
Force between two charges	F = k q₁q₂ / r²	1/r²	Vector
Electric field	E = kq / r² (also E = F/q)	1/r²	Vector
Electric potential	V = kq / r	1/r	Scalar
Potential energy	U = k q₁q₂ / r (also U = qV)	1/r	Scalar

The constant k is about 9 x 10⁹ N m²/C².

Sign rules

Like charges repel, opposite charges attract.
The field points away from positive charge and toward negative charge. A positive test charge feels force along E, a negative one feels force against E.
In U = k q₁q₂/r, keep the signs. Positive U means repulsive pair, negative U means bound (attractive) pair.
Positive charges move to lower potential on their own. Negative charges move to higher potential on their own. Either way, a free charge lowers its potential energy.

Scalars add simply, vectors need components

For several charges, total V is the plain sum of kq/r with signs. Total E needs vector addition. At the midpoint between two equal positive charges, E is zero (the fields cancel) but V is not zero (it is twice the single value). That contrast is a standard MCAT question.

Uniform field between plates

Between parallel plates, E = V/d is constant, and a charge moves with constant force F = qE. Equipotential lines are perpendicular to field lines, and no work is done moving a charge along an equipotential.

Worked example

Two charges, +2 uC and +3 uC, sit 0.30 m apart.

Force: F = (9 x 10⁹)(2 x 10^-6)(3 x 10^-6) / 0.30² = 0.054 / 0.09 = 0.60 N, repulsive.
Potential energy: U = 0.054 / 0.30 = 0.18 J.
If the distance doubles to 0.60 m, F falls to 0.15 N (one fourth) and U falls to 0.09 J (one half).

Common traps

Using 1/r² for potential. Potential and U go as 1/r.
Adding field magnitudes without direction. E is a vector.
Dropping signs in U. The sign tells repulsion or attraction.
Assuming zero field means zero potential. They are independent at a point.

One-minute recap

F and E: 1/r², vectors. V and U: 1/r, scalars. E = F/q, U = qV, E = V/d between plates. Positive charges fall toward lower V, negative toward higher V. No work along equipotentials.

Check yourself: 5-question mini-quiz

Try each question first, then click it to reveal the answer.

1. The distance between two point charges triples. By what factor does the force change?

It falls to 1/9, because force scales as 1/r squared.

2. The distance between two point charges triples. By what factor does the potential energy change?

It falls to 1/3, because U scales as 1/r.

3. At the midpoint between two equal positive charges, what are E and V?

E is zero because the fields cancel, but V is nonzero because potentials are scalars that add.

4. A negative charge is released in an electric field. Does it move toward higher or lower potential?

Higher potential, which lowers its potential energy.

5. A charge moves along an equipotential line. How much work does the field do?

Zero, because the potential difference is zero.

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Written by StudyStore from standard scientific facts. Study aid only. Always check values against the reference your exam provides.