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

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

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.