Coulomb's Law

Coulomb's Law quantifies the electrostatic force between two charges:

Fe = kq1q2/r²

where Fe is the magnitude of the electrostatic force between the two charges, k is Coulomb's constant, q1 and q2 are the magnitudes of the two charges, and r is the distance between the charges.

This equation tells us the force between two charges is directly proportional to the product of their magnitudes, and inversely proportional to the square of the distance between them. Double the distance between two charges, and the force drops to a quarter of its original value — not half.

Coulomb's Law applies only to charges at rest, which is why it's central to electrostatics. The resulting force can be attractive or repulsive depending on the charges' signs: opposite charges pull toward each other, like charges push apart.

MCAT Callout — Worked Example: Force Between Two Point Charges: Two charges, q1 = +2×10⁻⁶ C and q2 = −3×10⁻⁶ C, are separated by 0.5 m. The force between them is Fe = (8.99×10⁹)(2×10⁻⁶)(3×10⁻⁶) / (0.5)² ≈ 0.216 N. Since the charges have opposite signs, this force is attractive.

Key Takeaways

• Coulomb's Law: Fe = kq1q2/r² — force is proportional to the product of charge magnitudes, inversely proportional to the square of the distance.

• Coulomb's constant k = 8.99×10⁹ N·m²/C², related to the permittivity of free space by k = 1/(4πε0).

• With more than two charges, find the net force by vector-summing the pairwise Coulomb forces (superposition).

• The electric field (E = Fe/q = kQ/r²) is a property of a source charge's surrounding space, distinct from the force felt by any particular test charge.

• Electric field lines point outward from positive charges, inward toward negative charges, never cross, and their density shows field strength.

Coulomb's Constant and Permittivity of Free Space

Coulomb's constant, k, has a value of k = 8.99 × 10⁹ N·m²/C².

Coulomb's constant can be derived from the permittivity of free space, denoted ε0 (epsilon-naught): k = 1/(4πε0).

The permittivity of free space describes how charges interact in a vacuum, and it reappears frequently across electrostatics equations — including the electric field of a dipole discussed later in this chapter.

Multiple Charges: Superposition of Forces

Coulomb's Law still applies when more than two charges are present, but the approach must be extended. When multiple charges act on a single charge, the net electrostatic force on that charge is the vector sum of the forces exerted by each of the other charges individually.

In practice, this means calculating the force between each pair of charges separately using Coulomb's Law, then summing those forces — keeping track of both magnitude and direction for each.

The Electric Field

Just as every mass creates a gravitational field, every electric charge sets up a surrounding electric field — a region of space where it can exert a force on other charges. This field exists even when no other charge happens to be present; it's a property of the charge itself, the same way a gravitational field surrounds a planet regardless of whether another object is in that field.

An electric field becomes apparent when another charge enters it. The charge creating the field is the source charge (Q); the charge that enters the field and experiences a force is the test charge (q). If the two charges have opposite signs, the force is attractive, drawing the test charge toward the source; if they share the same sign, the force is repulsive, pushing the test charge away.

The electric field's magnitude is given by:

E = Fe/q = kQ/r²

where E is the electric field strength, Fe is the electrostatic force, q is the test charge, k is Coulomb's constant, Q is the source charge, and r is the distance between the source charge and the point where the field is measured.

MCAT Callout — Force vs. Field: The electrostatic force (Fe) exists only between two specific charges. The electric field (E) is a property of a single source charge's surrounding space — it exists regardless of whether a test charge is there to feel it. The field is simply the force per unit test charge: E = Fe/q.

Electric Field Lines

Electric field lines are a visual tool for representing the direction and strength of an electric field. A few key properties govern how they're drawn and interpreted:

• Lines point in the direction of the electric field at any given point — they radiate outward from positive charges and point inward toward negative charges.

• Line density indicates field strength: the closer together the lines are, the stronger the field in that region.

• Field lines always start on positive charges and end on negative charges; the number of lines at a charge is proportional to that charge's magnitude.

• Field lines never cross — the electric field can't point in two directions or exert two different forces at the same point.

Between a positive and a negative charge, field lines curve from one charge to the other, illustrating attraction. Between two like charges, the lines curve away from each other, illustrating repulsion.

Common MCAT Mistakes

• Treating the inverse-square relationship as inverse-linear. Halving the distance between two charges quadruples the force, not doubles it — the r² in the denominator means force scales with the square of distance, not distance itself.

• Ignoring the sign of the charges. The magnitude of Fe tells you how strong the force is, but the signs of q1 and q2 tell you the direction: opposite signs attract, like signs repel. Dropping the signs loses the direction of the force.

• Conflating force and field. The electrostatic force (Fe) only exists between two specific charges. The electric field (E) is a property of a single source charge's surrounding space, present whether or not a test charge is there to feel it.

• Adding force magnitudes directly instead of vector-summing. With more than two charges present, the net force on a charge is the vector sum of the pairwise Coulomb forces — magnitude and direction both matter, not just magnitude.

MCAT-Style Concept Check

Question: Two point charges exert an electrostatic force of 12 N on each other when separated by a distance r. If the distance between the charges is tripled to 3r, with the charge magnitudes unchanged, what is the new force between them?

• A) 36 N

• B) 4 N

• C) 1.33 N

• D) 0.44 N

Answer: C

Explanation: By Coulomb's Law, Fe = kq1q2/r², so force is inversely proportional to the square of the distance. Tripling the distance increases r² by a factor of 3² = 9, so the force drops to 1/9 of its original value: 12 N / 9 ≈ 1.33 N.

FAQ

What is Coulomb's Law?

Coulomb's Law states that the electrostatic force between two charges (Fe = kq1q2/r²) is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. It applies to charges at rest and can produce either an attractive or repulsive force depending on the charges' signs.

What is Coulomb's constant?

Coulomb's constant, k, equals 8.99×10⁹ N·m²/C². It can also be derived from the permittivity of free space (ε0) using k = 1/(4πε0), a relationship that reappears throughout electrostatics.

How do you find the net force when more than two charges are present?

Calculate the Coulomb force between each pair of charges separately, then find the vector sum of those individual forces — this is called superposition. Both magnitude and direction matter for each pairwise force.

What's the difference between electric force and electric field?

Electric force (Fe) exists only between two specific charges. Electric field (E) is a property of a single source charge's surrounding space — it exists regardless of whether a test charge is present. The field equals the force per unit test charge: E = Fe/q.