Coulomb's Law in AP Physics C: Electricity and Magnetism

In AP Physics C: E&M, Coulomb's Law says the electrostatic force between two point charges is F = k|q₁q₂|/r², where k = 1/(4πε₀). The force acts along the line joining the charges. It is attractive for opposite charges and repulsive for like charges.

What is Coulomb's Law?

Coulomb's Law is the rule for how hard two charges push or pull on each other. The magnitude is F = k|q₁q₂|/r². Here q₁ and q₂ are the charges, r is the distance between them, and k is Coulomb's constant (about 8.99 × 10⁹ N·m²/C²). You'll also see k written as 1/(4πε₀), where ε₀ is the permittivity of free space. Double one charge and the force doubles. Double the distance and the force drops to one quarter. That inverse-square drop is the part people forget.

In Physics C, treat Coulomb's Law as a vector law, not just a magnitude formula. The force points along the line connecting the two charges. Like charges repel and opposite charges attract. Strictly speaking, the law applies to point charges. When you have several charges, you find each pairwise force separately and add them as vectors (superposition). When charge is spread out over a rod, ring, or disk, you break it into tiny pieces dq and integrate. Think of it this way. Coulomb's Law is the "atom" of electrostatics, and every bigger charge setup gets built by adding up a lot of tiny Coulomb's Law pieces.

Why Coulomb's Law matters in AP® Physics C: E&M

Coulomb's Law lives in Topic 8.1, Electric Charge and Electric Force, at the very start of Unit 8: Electric Charges, Fields, and Gauss's Law. It's the foundation for the rest of the unit. The electric field is just Coulomb force per unit charge. Gauss's Law is the same physics written in a form that's easier to use when the setup is symmetric. Electric potential energy in Unit 9 comes from integrating the Coulomb force over distance. If your Coulomb's Law skills are shaky (vector components, superposition, setting up dq integrals), the field and potential problems in Units 8-9 get much harder. It also ties into the course's big idea that charge interactions explain everyday forces, like the normal force from a table.

How Coulomb's Law connects across the course

Electric Field (Unit 8)

The electric field is Coulomb's Law with the test charge divided out, so E = F/q = kQ/r² for a point charge. The field tells you how strong the push would be on any charge placed at that spot, before you actually put one there.

Gauss's Law (Unit 8)

Gauss's Law and Coulomb's Law describe the same physics. Draw a spherical Gaussian surface around a point charge and Gauss's Law hands you E = Q/(4πε₀r²), which is exactly Coulomb's result. Use Coulomb's Law (and integration) for asymmetric setups, and switch to Gauss's Law when spheres, cylinders, or planes make the math easy.

Electric Potential Energy and Potential (Unit 9)

Integrate the Coulomb force over distance and you get potential energy U = kq₁q₂/r. Notice that it's 1/r, not 1/r². Energy is a scalar, so adding up several charges is much easier than adding Coulomb force vectors.

Conductors and Capacitors (Unit 10)

Charges on a conductor spread out because they repel each other according to Coulomb's Law. They keep moving until the field inside the conductor is zero. That's why excess charge ends up on the surface.

Is Coulomb's Law on the AP® Physics C: E&M exam?

Coulomb's Law shows up mostly in three ways.

• Vector superposition. Several charges sit at the corners of a shape, like an equilateral triangle with side L, and you find the net force on one of them. You break each Coulomb force into components and add them, using the symmetry to cancel pieces.
• Force balance. A common setup has charged particles where electrostatic repulsion or attraction must balance gravity. You'll solve for a required charge q in terms of m, g, L, and k (or ε₀). You do this by setting the net Coulomb force equal to mg and paying attention to direction.
• Conceptual modeling. Some questions ask why the normal force, or a collision between charged blocks, is often modeled with contact forces or a coefficient of restitution instead of direct Coulomb calculations. The answer is that you'd have to sum an enormous number of atomic-scale interactions, so a simpler macroscopic model works better.

No released FRQ has used the phrase "Coulomb's Law" verbatim. Even so, it sits under most Unit 8 FRQs, especially ones where you integrate dq over a rod or ring to find a force or field. Expect to write the law symbolically, justify directions, and check limiting cases (for example, a far-away distribution should act like a point charge).

Coulomb's Law vs Newton's Law of Universal Gravitation

Both are inverse-square laws with the same shape: a constant times a product, divided by r². The differences matter, though. Gravity uses masses, is always attractive, and is extremely weak (G ≈ 6.67 × 10⁻¹¹ N·m²/kg²). Coulomb's Law uses charges, which can be positive or negative, so the force can attract or repel. It's also enormously stronger (k ≈ 8.99 × 10⁹ N·m²/C²). In problems where both act, gravity usually only matters for macroscopic masses or when the charges are tiny.

Key things to remember about Coulomb's Law

• Coulomb's Law gives the force between two point charges as F = k|q₁q₂|/r², with k = 1/(4πε₀) ≈ 8.99 × 10⁹ N·m²/C².

• The force acts along the line joining the charges, repelling like charges and attracting opposite charges.

• Doubling the distance between two charges cuts the force to one quarter, because the force depends on 1/r².

• With multiple charges, you find each pairwise force separately and add them as vectors using components and symmetry.

• For continuous charge distributions, you treat each small piece dq as a point charge and integrate.

• Coulomb's Law is the starting point for electric field (E = F/q), Gauss's Law, and electric potential energy (U = kq₁q₂/r).

Frequently asked questions about Coulomb's Law

What is Coulomb's Law in AP Physics C E&M?

Coulomb's Law states that the electric force between two point charges is F = k|q₁q₂|/r², directed along the line between them. It's introduced in Topic 8.1 and is the base for electric fields and Gauss's Law in Unit 8.

Does Coulomb's Law work for objects that aren't point charges?

Not directly. It strictly applies to point charges, or to spherically symmetric charges viewed from outside. For rods, rings, or disks, you split the object into tiny pieces dq, apply Coulomb's Law to each piece, and integrate.

Is the normal force actually an electric force?

Yes, at the atomic level. It comes from electrons in the book and table repelling each other. Physicists still treat it as a separate contact force, because summing trillions of Coulomb interactions is impractical and a macroscopic model gives the same result.

How is Coulomb's Law different from Newton's law of gravitation?

Both are inverse-square laws, but gravity acts between masses and only attracts. The Coulomb force acts between charges and can attract or repel. Coulomb's constant (about 8.99 × 10⁹) dwarfs G (about 6.67 × 10⁻¹¹), so electric forces are usually far stronger.

What's the difference between Coulomb's Law and the electric field?

Coulomb's Law gives the force between two specific charges. The electric field is that force per unit test charge, E = F/q. For a point charge that works out to E = kQ/r², and it exists whether or not a second charge is present.