---
title: "Coulomb's Constant — AP Physics C: E&M Definition & Guide"
description: "Coulomb's constant (k ≈ 8.99 × 10⁹ N·m²/C²) sets the strength of the electrostatic force and appears in field, potential, and energy equations across Unit 1."
canonical: "https://fiveable.me/ap-physics-c-e-m/key-terms/coulombs-constant"
type: "key-term"
subject: "AP Physics C: E&M"
unit: "Unit 1"
---

# Coulomb's Constant — AP Physics C: E&M Definition & Guide

## Definition

Coulomb's constant, k ≈ 8.99 × 10⁹ N·m²/C², is the proportionality constant in Coulomb's law that converts charges and distance into electrostatic force; it equals 1/(4πε₀) and shows up in nearly every electrostatics equation in AP Physics C: E&M, from point-charge fields to potential energy.

## What It Is

Coulomb's constant, written as k, is the number that makes [Coulomb's law](/ap-physics-c-e-m/key-terms/coulombs-law "fv-autolink") work. The law says the electrostatic force between two point charges is F = kq₁q₂/r². The charges and the distance tell you the *shape* of the relationship, but k sets the actual *strength*. Its value is about 8.99 × 10⁹ N·m²/C², which is huge. That enormous size is why electric forces dominate at the atomic scale even though individual charges are tiny.

In [AP Physics C: E&M](/ap-physics-c-e-m "fv-autolink") you'll also see k written as 1/(4πε₀), where ε₀ is the [permittivity of free space](/ap-physics-c-e-m/unit-10/4-dielectrics/study-guide/94aiEgDjuJxhK3Px "fv-autolink"). They're the same physical content in two different costumes. The k form is convenient for quick point-charge calculations. The ε₀ form is what you'll use once Gauss's law enters the picture, because flux equations are built around ε₀. Both forms are on the AP equation sheet, so you never need to memorize the value, but you do need to recognize k in every equation it lives in: force (kq₁q₂/r²), electric field of a point charge (kq/r²), electric potential (kq/r), and electric potential energy (kq₁q₂/r).

## Why It Matters

Coulomb's constant lives in Unit 1 (Electrostatics) and maps to Topic 1.5, Other [Charge](/ap-physics-c-e-m/unit-10/2-redistribution-of-charge-between-conductors/study-guide/3zelmsMupFfJh7VP "fv-autolink") Distributions - Fields & Potentials. When you calculate the field or potential of a ring, arc, or line of charge by [integration](/ap-physics-c-e-m/key-terms/integration "fv-autolink"), every infinitesimal piece contributes dE = k dq/r² or dV = k dq/r, so k is the constant you carry through the entire integral. It's also the bridge constant of the whole unit. The same k connects force, field, potential, and potential energy, which is exactly the web of relationships Topic 1.5 asks you to use. If you can spot where k sits in an equation, you can usually tell whether you're dealing with a force-like quantity (1/r² dependence) or a potential-like quantity (1/r dependence), and that distinction is one of the most common things E&M questions probe.

## Connections

### [Electrostatic Force (Unit 1)](/ap-physics-c-e-m/key-terms/electrostatic-force)

Coulomb's law is where k was born. F = kq₁q₂/r² is the foundational equation of the unit, and everything else in electrostatics is a repackaging of this force per unit charge or as stored energy.

### Electric Field (Unit 1)

Divide Coulomb's law by a [test charge](/ap-physics-c-e-m/unit-8/3-electric-fields/study-guide/7Nyjo6HcMeSSkleV "fv-autolink") and you get E = kq/r², the field of a point charge. In Topic 1.5 you integrate k dq/r² over rings and rods, so k rides along in every charge-distribution integral.

### Electric Potential Energy (Unit 1)

U = kq₁q₂/r uses the same constant but only one power of r, because potential energy is the integral of force over distance. Watching the exponent next to k (r² for force and field, r for potential and energy) is a fast error-check on the exam.

### [Ring of Charge (Unit 1)](/ap-physics-c-e-m/key-terms/ring-of-charge)

The classic ring-of-charge derivation pulls k outside the integral since it's a constant, leaving you to integrate over dq. Recognizing that k (and often r) factors out is what makes these Topic 1.5 integrals doable under time pressure.

## On the AP Exam

You won't be asked "what is Coulomb's constant?" directly. Instead, k is the constant you must use correctly while doing something else. On multiple choice, it shows up in symbolic answer choices, and a wrong exponent on r next to k is a favorite distractor. On free response, derivation problems (find E or V from a ring, rod, or arc of charge) require you to set up an integral with k dq in it, pull k outside, and keep it through to the final boxed expression. Also expect to switch between k and 1/(4πε₀) forms, since Gauss's law problems are written in terms of ε₀ while point-charge formulas usually use k. Both constants are given on the equation sheet, so the skill being tested is fluent use, not recall.

## Coulomb's constant vs Permittivity of free space (ε₀)

These are two ways of writing the same physics, related by k = 1/(4πε₀). Coulomb's constant k (≈ 8.99 × 10⁹ N·m²/C²) is large and multiplies things in point-charge formulas. The permittivity ε₀ (≈ 8.85 × 10⁻¹² C²/N·m²) is tiny and sits in the denominator of Gauss's law (Φ = Q/ε₀). If you mix them up, your answer will be off by a factor of roughly 10²⁰, which is usually an instant red flag. Rule of thumb for the exam: point charges and discrete sums lean on k, while Gauss's law and capacitance lean on ε₀.

## Key Takeaways

- Coulomb's constant k is approximately 8.99 × 10⁹ N·m²/C² and sets the strength of the electrostatic force in Coulomb's law, F = kq₁q₂/r².
- k equals 1/(4πε₀), so it and the permittivity of free space are interchangeable; use whichever form matches the equation you're working with.
- The same k appears in force (kq₁q₂/r²), field (kq/r²), potential (kq/r), and potential energy (kq₁q₂/r), with the power of r telling you which quantity you're looking at.
- In Topic 1.5 integration problems, k is a constant and comes outside the integral, so set up ∫k dq/r² as k∫dq/r² before integrating.
- Both k and ε₀ are printed on the AP equation sheet, so the exam tests whether you can use them correctly, not whether you memorized them.

## FAQs

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

It's the proportionality constant k ≈ 8.99 × 10⁹ N·m²/C² in Coulomb's law, F = kq₁q₂/r². It converts the product of two charges divided by distance squared into an actual force in newtons.

### Do I have to memorize Coulomb's constant for the AP exam?

No. Both k and ε₀ are provided on the AP Physics C equation sheet. What you do need is to use k correctly in symbolic derivations and know it equals 1/(4πε₀).

### Is Coulomb's constant the same as the permittivity of free space?

No, but they're directly related by k = 1/(4πε₀). k is about 8.99 × 10⁹ and ε₀ is about 8.85 × 10⁻¹², so confusing them changes your answer by roughly 20 orders of magnitude.

### When should I use k versus ε₀ in a problem?

Use k for point-charge formulas like F = kq₁q₂/r², E = kq/r², and V = kq/r. Use ε₀ when the equation is built around it, like Gauss's law (Φ = Q/ε₀) or [capacitance](/ap-physics-c-e-m/key-terms/capacitance "fv-autolink"). Converting between them with k = 1/(4πε₀) is fair game on the exam.

### Why is Coulomb's constant so big?

Because a coulomb is an enormous amount of charge compared to anything in everyday life. The large value of k means even microcoulomb-scale charges produce sizable forces, which is why [electrostatic forces](/ap-physics-c-e-m/key-terms/electrostatic-force "fv-autolink") dominate gravity at the atomic scale.

## Related Study Guides

- [1.5 Other Charge Distributions - Fields & Potentials](/ap-physics-c-e-m/unit-1/other-charge-distributions-fields-potentials/study-guide/hdmmpVBcGfnIHm4HoA0j)

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