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

In AP Physics C: E&M, Coulomb's constant (k ≈ 8.99×10⁹ N·m²/C²) is the proportionality constant in Coulomb's law, F = k|q₁q₂|/r². It sets how strong the electrostatic force is between point charges and equals 1/(4πε₀).

What is Coulomb's constant?

Coulomb's constant, written as k, is the number that turns Coulomb's law from a proportionality into an actual equation. Coulomb's law says electric force grows with the product of the charges and shrinks with the square of the distance. The constant k tells you how much force you actually get: F = k|q₁q₂|/r², with k ≈ 8.99×10⁹ N·m²/C² (you'll often round to 9×10⁹).

That huge value is the whole story of why electricity is so strong. Put two 1 C charges 1 m apart and they push on each other with about 9 billion newtons. The constant also has a second identity you need to know: k = 1/(4πε₀), where ε₀ is the permittivity of free space (≈ 8.85×10⁻¹² C²/(N·m²)). Same physics, two ways of writing it. Early equations in the course usually use k, and Gauss's law and capacitor formulas usually use ε₀.

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

Coulomb's constant lives in Topic 8.1, Electric Charge and Electric Force, in Unit 8: Electric Charges, Fields, and Gauss's Law. It's the first constant you meet in E&M, and it never goes away. Every point-charge formula in Units 8 and 9 carries a k: force, field, potential energy, and potential. Once Gauss's law shows up, k quietly changes costume into 1/(4πε₀), and that ε₀ then drives capacitance in Unit 10. If you understand k and its link to ε₀, you can move between all of these equations without memorizing them as separate facts. For the full treatment of charge and Coulomb's law, see the 8.1 topic guide.

How Coulomb's constant connects across the course

Coulomb's Law (Unit 8)

Coulomb's law gives you the shape of the electric force (product of charges over distance squared), and k gives you the size. Think of the law as the recipe and k as the conversion that turns coulombs and meters into newtons.

Electric Field of a Point Charge (Unit 8)

Divide Coulomb's law by a test charge and you get E = kq/r². The field is just the force per unit charge, so the same k carries over unchanged.

Permittivity of Free Space and Gauss's Law (Units 8, 10)

Gauss's law is written with ε₀, not k, because the 4π naturally cancels with the area of a sphere. That's why k = 1/(4πε₀) is written that way. The same ε₀ then shows up in the parallel-plate capacitor, C = ε₀A/d.

Electric Potential and Potential Energy (Unit 9)

Integrate the Coulomb force over distance and you get U = kq₁q₂/r, then V = kq/r. The constant stays put, but the distance drops from r² to r, which is the easiest way to tell these formulas apart.

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

You'll rarely be asked to define k. You'll be asked to use it. Common moves on multiple-choice and free-response questions:

• Plug-and-chug Coulomb's law. Find the force between two point charges given in μC or nC. Watch your prefixes (μ = 10⁻⁶, n = 10⁻⁹).
• Solve for k itself. Some questions give you the force, charges, and separation and ask what k would be in that scenario. Rearrange to k = Fr²/|q₁q₂|.
• Balance forces. A classic setup has a charged sphere hanging from a string with a second charge directly below it. When the tension goes to zero, the electric force kq₁q₂/r² equals mg, and you solve for r.
• Compare electric and gravitational force. Questions pair k with G to show that electric forces between small charged objects can dwarf gravity, since k is about 10²⁰ times larger than G.
• Spot the trap. If one object is neutral, as in a charged Earth and a neutral Sun, there's no Coulomb force between them no matter how big k is. q = 0 makes the whole product zero.

No released FRQ has used the term "Coulomb's constant" verbatim, but k (or 1/(4πε₀)) is built into nearly every derivation involving point charges, fields, and potentials.

Coulomb's constant vs Permittivity of Free Space (ε₀)

These aren't two different constants describing two different things. They describe the same thing. k = 1/(4πε₀), so k ≈ 8.99×10⁹ N·m²/C² and ε₀ ≈ 8.85×10⁻¹² C²/(N·m²). The trap is mixing them up in a formula. A big k goes in the numerator (F = kq₁q₂/r²). A tiny ε₀ goes in the denominator (E = q/(4πε₀r²) or E = σ/ε₀). If your answer comes out off by about 10²¹, you probably swapped them.

Key things to remember about Coulomb's constant

• Coulomb's constant k is about 8.99×10⁹ N·m²/C², and it sets the strength of the electrostatic force in Coulomb's law.

• The constant k is the same thing as 1/(4πε₀), so any equation written with k can be rewritten with the permittivity of free space.

• The same k appears in the point-charge formulas for force, field, potential energy, and potential, so learn it once and reuse it everywhere.

• Force and field formulas divide by r², while potential energy and potential formulas divide by r, and k stays the same in all four.

• Because k is so much larger than G, electric forces between small charged objects are usually far stronger than the gravitational forces between them.

• If either charge is zero, the Coulomb force is zero, no matter how large k is.

Frequently asked questions about Coulomb's constant

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

Coulomb's constant k is the proportionality constant in Coulomb's law, F = k|q₁q₂|/r², with a value of about 8.99×10⁹ N·m²/C². It's introduced in Topic 8.1 and reappears in every point-charge formula in Units 8 and 9.

Is Coulomb's constant the same as ε₀?

No, but they're directly linked. k = 1/(4πε₀), so k is huge (≈ 8.99×10⁹) and ε₀ is tiny (≈ 8.85×10⁻¹²). They express the same physics in two forms, and Gauss's law and capacitor equations use ε₀.

How is Coulomb's constant different from the gravitational constant G?

Both constants play the same role in inverse-square laws, but k ≈ 8.99×10⁹ N·m²/C² while G ≈ 6.67×10⁻¹¹ N·m²/kg². That gap of roughly 20 orders of magnitude is why electric forces usually dominate gravity for small charged objects.

Does Coulomb's constant change depending on the problem?

In real-world problems, no. k is about 9×10⁹ N·m²/C² in a vacuum (and essentially in air). Some practice questions describe a hypothetical scenario and ask you to solve for k using k = Fr²/|q₁q₂|, which is a test of rearranging Coulomb's law, not a change in physics.

Do I need to memorize the value of Coulomb's constant for the AP exam?

You should know it's about 9×10⁹ N·m²/C² and that it equals 1/(4πε₀), but the value is provided on the AP Physics C equation and constants sheet. What really matters is knowing which formulas use it and keeping your unit prefixes straight.