Physics · Fields & Magnetism · Grade 9-12 · 5 min read

Coulomb's Law

⚡ In one breath

Coulomb's Law gives the electrostatic force between two point charges: $F = kq_1q_2/r^2$ , larger for larger charges and falling off as the square of the separation.

📐 The formula

F = k\frac{|q_1||q_2|}{r^2}

where

k \approx 8.99 \times 10^9

N m

^2

^2

Orient

The one-line idea, why it matters, and the intuition.

Section 1

Quick Answer

Coulomb's Law gives the electrostatic force between two point charges: $F = kq_1q_2/r^2$ , larger for larger charges and falling off as the square of the separation. Recognize it when the problem names two charges, gives the distance between them, and asks for the force. The key check is the inverse-square distance and that the answer is a force in newtons — not a field (Electric Field) or a voltage (Electric Potential). Measure $r$ centre to centre before computing.

Section 2

Why This Matters

Coulomb's Law gives students a way to explain non-contact forces and energy changes. It connects electricity, magnetism, gravitation, induction, motors, generators, and orbital motion through a shared spatial model.

Section 3

Intuitive Explanation

Coulomb's Law answers one specific question: how hard do two charges push or pull on each other across empty space? Like charges repel, opposite charges attract, and the strength depends on both charge values multiplied together and on the distance squared in the denominator. That last part is the feature to lock onto — double the separation and the force drops to a quarter, not a half.

The surest way to spot it is to count the charges. Coulomb's Law always involves two of them, and the answer it produces is a single force pointing along the line joining them. If the problem instead hands you one source charge and asks what a stray test charge would feel anywhere nearby, you have crossed into Electric Field territory; if it asks for the work or voltage at a point, that is Electric Potential.

Once you have confirmed it is a two-charge force question, plug the charges and the centre-to-centre distance into $F = kq_1q_2/r^2$ . A common trap is using the gap between sphere surfaces instead of the distance between their centres — for point charges and uniform spheres, the centre distance is what the law requires.

Core idea

Coulomb's Law starts by naming the source, the object affected, and how the field or potential changes through space.

Recognize

The cues that signal this concept and how to distinguish it from look-alikes.

Section 4

When to Use

Use Coulomb's Law when a problem names two point charges (or charged spheres treated as points), gives their charges and the separation between them, and asks for the electrostatic force one exerts on the other. The recognition cue is two charges plus a distance, with a force as the answer. The nearest confusion is Electric Field, which uses a single source charge to describe what a test charge would feel at any point in space; if only one charge is named, reach for the field instead. Do not invoke Coulomb's Law just because charge appears in the wording — first confirm there are two interacting charges and that the question wants the force between them.

Pro tip

Ask: Am I using a field or potential to explain how one object influences another across space?

Section 5

How to Recognize It

Coulomb's Law is the force law between two point charges. Before using it, check that the problem really gives you two charges and a separation and wants the force between them.

Are there exactly two charged objects (point charges or uniform spheres) and a distance $r$ stated between them?

Two charges plus a separation is the structural signature of Coulomb's Law. If only one charge is described, or no separation is given, a different electrostatics concept fits better.
Is the quantity asked for a force (in newtons), not a field, a voltage, or an energy?

Coulomb's Law outputs a force between the two charges. If the prompt wants the field strength or the potential at a point, you want Electric Field or Electric Potential instead.
If the distance were doubled, do you expect the answer to drop to one quarter?

Yes confirms the inverse-square dependence that defines Coulomb's Law. If the relationship is supposed to be linear or constant, this is not the right law.
Is the nearest sibling really Electric Field — a single source charge and you want what a test charge would feel anywhere around it?

If you only have one source and want $E = kq/r^2$ as a property of space, that is Electric Field. Coulomb's Law needs both interacting charges named.
Are you measuring distance between the centres of the charges, not between their surfaces?

Coulomb's Law uses centre-to-centre $r$ for point charges and uniform spheres. If the problem only gives a surface gap, you must reconstruct the centre distance before the law applies.

Exam shortcut

Decide which concept the problem needs

Use Coulomb's Law

Likely Coulomb's Law: the problem names two point charges (or charged spheres), gives their charge values and the separation $r$ , and asks for the force between them — and the structure answers "Am I finding the force one charge exerts on another?" with yes. In that case write $F = kq_1q_2/r^2$ and note the centre-to-centre distance before plugging in.

Use another concept

Unlikely Coulomb's Law: if the task only asks about the charge itself (counting or conserving charge) it is Electric Charge; if it asks for the field at a point produced by a charge it is Electric Field; if it asks for voltage or potential energy at a location it is Electric Potential. Switch to that concept before you compute.

Section 6

Coulomb's Law vs Electric Charge vs Electric Field vs Electric Potential

These get mixed up because all four are electrostatics built on charge. The deciding cue is what the problem hands you and asks for: Coulomb's Law needs two charges, a separation, and wants a force, while the other rows fit when you count charge, find the field at a point, or find the voltage there.

Coulomb's Law vs Electric Charge vs Electric Field vs Electric Potential
Type	Meaning	Key test	Formula	Example
Coulomb's Law	Use when two point charges (or charged spheres) are named with their charges and the separation, and the question asks for the force one exerts on the other.	Are there two charges a distance apart, and is a force the answer?	$F=k\dfrac{\|q_1\|\|q_2\|}{r^2}$	Charges of 2 μC and 3 μC separated by 0.50 m exert about 0.22 N on each other via $F=kq_1q_2/r^2$ .
Electric Charge	Use when the question is about the charge itself — counting it, conserving it, or how an object becomes charged — not a force.	Is the quantity the charge alone, with no second charge or distance?	$q$ in coulombs	Rubbing a balloon on hair transfers electrons, leaving the balloon negatively charged.
Electric Field	Use when a single source charge is given and you want what a test charge would feel at a point in space — a field, not a force between two named charges.	Is only one charge named, with the field at a location wanted?	$E=\dfrac{kQ}{r^2}$	Hold a charged balloon near scraps of paper — they jump toward it, responding to the field around it.
Electric Potential	Use when the answer is a voltage or potential energy per unit charge at a location, not a force.	Is the answer a voltage or energy at a point rather than a force?	$V=\dfrac{kQ}{r}$	A point 1 m from a +1 μC charge sits at about 9000 V.

Coulomb's Law

Meaning: Use when two point charges (or charged spheres) are named with their charges and the separation, and the question asks for the force one exerts on the other.
Key test: Are there two charges a distance apart, and is a force the answer?
Formula: $F=k\dfrac{|q_1||q_2|}{r^2}$
Example: Charges of 2 μC and 3 μC separated by 0.50 m exert about 0.22 N on each other via $F=kq_1q_2/r^2$ .

Electric Charge

Meaning: Use when the question is about the charge itself — counting it, conserving it, or how an object becomes charged — not a force.
Key test: Is the quantity the charge alone, with no second charge or distance?
Formula: $q$ in coulombs
Example: Rubbing a balloon on hair transfers electrons, leaving the balloon negatively charged.

Electric Field

Meaning: Use when a single source charge is given and you want what a test charge would feel at a point in space — a field, not a force between two named charges.
Key test: Is only one charge named, with the field at a location wanted?
Formula: $E=\dfrac{kQ}{r^2}$
Example: Hold a charged balloon near scraps of paper — they jump toward it, responding to the field around it.

Electric Potential

Meaning: Use when the answer is a voltage or potential energy per unit charge at a location, not a force.
Key test: Is the answer a voltage or energy at a point rather than a force?
Formula: $V=\dfrac{kQ}{r}$
Example: A point 1 m from a +1 μC charge sits at about 9000 V.

Apply

Worked examples and the mistakes most students make.

Section 7

Formula & Notation

F = k\frac{|q_1||q_2|}{r^2}

where

k \approx 8.99 \times 10^9

N m

^2

^2

Coulomb's law in vector form is

\vec{F}_{12} = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}\hat{r}_{12}

, where

\hat{r}_{12}

points from charge 1 to charge 2. The constant

k = 1/(4\pi\epsilon_0) \approx 8.99 \times 10^9

N·m²/C².

How to read it: $F$ is the force in newtons, $q_1$ and $q_2$ are the charges in coulombs, $r$ is the separation in metres, $k$ is Coulomb's constant, and $\epsilon_0$ is the permittivity of free space.

Section 8

Worked Examples

Example 1 — Recognize the model

Easy

Problem

A class observes this situation: a charged object is brought near another object and the second object experiences a force without touching it. How should a student decide whether Coulomb's Law is the right model?

Solution

Identify the system.

Physics models apply to a chosen object, region, circuit, wave, fluid, or particle. Without the system, the quantities have no target.
List the quantities or interactions that matter.

Coulomb's Law is useful when the problem asks for a field, force, potential, flux, or induced effect with direction and units stated when needed.
Apply the recognition test: Am I using a field or potential to explain how one object influences another across space?

This separates coulomb's law from contact force and potential difference.
Write the answer form before solving.

Knowing whether the result needs units, direction, a boundary condition, or a before-and-after comparison prevents formula guessing.

Answer

Use Coulomb's Law only if the problem is asking for a field, force, potential, flux, or induced effect with direction and units stated when needed and the system passes the recognition test. Otherwise, choose the nearby model that better matches the system.

Takeaway: Model choice comes before calculation. The same numbers can belong to different physics ideas depending on the system boundary.

Example 2 — Avoid the formula trap

Standard

Problem

A student says, "This problem contains the word field, so I should use coulomb's law." Explain why that shortcut is risky.

Solution

Treat the word as a clue, not proof.

Physics vocabulary overlaps across models, so one word cannot choose the law by itself.
Check whether the object and interaction match Coulomb's Law.

The physical structure decides the model.
Compare with Contact force and Potential difference.

Contact forces require touching; field forces can act across space. Potential difference compares two points; a field describes the local influence in space.
State what the final result would mean.

If the final result would not mean a field, force, potential, flux, or induced effect with direction and units stated when needed, the model is probably wrong.

Answer

The shortcut is risky because field can appear in several related models. The student must first show that the system answers "Am I using a field or potential to explain how one object influences another across space?" with yes.

Takeaway: A physics formula is a model written compactly, not a keyword response.

Example 3 — Write the physical conclusion

Application

Problem

After solving a Coulomb's Law problem, a student writes only a number. What should be added to make the answer physically meaningful?

Solution

Attach units and direction when relevant.

Units and direction identify the quantity. A bare number often cannot distinguish related physics ideas.
Name the system and conditions.

The result may apply only for a chosen object, circuit path, medium, reference frame, or time interval.
Connect the result to the observation.

The final sentence should explain what the number says about the physical behavior.
Mention the assumption if the model is idealized.

Assumptions like no friction, closed system, constant speed, ideal gas, or no air resistance control when the result is valid.

Answer

A complete answer should say what the result means for the chosen system, include the correct units or direction, and state any condition needed for the coulomb's law model to apply.

Takeaway: The final explanation is part of the physics, not an optional sentence after the math.

Section 9

Common Mistakes

Common slip-up

Forgetting the inverse-square dependence

The right idea

halving the distance quadruples the force, not just doubles it. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.

Common slip-up

Using the distance between the surfaces of two charged spheres instead of the distance between their centres

The right idea

Coulomb's law uses centre-to-centre distance for point charges and uniform spheres. - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.

Common slip-up

Dropping the sign of the charges and getting the force direction wrong

The right idea

like charges repel (positive force away) and unlike charges attract (force toward each other). - Fix this by naming the system, checking "Am I using a field or potential to explain how one object influences another across space?", and attaching units or direction to the final statement.

Common slip-up

Using coulomb's law from a keyword alone

The right idea

Signal words like field, charge, magnet only point to a possible model; the system must match too.

Practice

Try it, then see where this concept fits in the path.

Section 10

Mini Practice

Try these on your own. Tap Reveal when you want to check.

What clue tells you this is Coulomb's Law: two small charged spheres, +4 μC and −6 μC, are held 0.30 m apart, and you must find the force between them?

Hint: Count the charges and look for a distance.
Why is this a contrast case instead of Coulomb's Law: a single +5 μC charge sits alone, and you must find what a test charge would feel 0.10 m away?

Hint: How many charges are actually named?
Two charges are at force $F$ . If you halve the distance between them while keeping the charges fixed, what is the new force?

Hint: Inverse-square, not inverse.
Why is finding the voltage 1 m from a +1 μC charge a contrast case rather than Coulomb's Law?

Hint: What kind of quantity is the answer?
Two charged metal spheres each of radius 2 cm have their centres 10 cm apart. What distance $r$ goes into Coulomb's Law, and why?

Hint: Surfaces or centres?
What single recognition question separates Coulomb's Law from electric field, charge, and potential?

Hint: Two charges and a force.

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Section 11

Frequently Asked Questions

What is Coulomb's Law in simple terms?

Coulomb's Law gives the electrostatic force between two point charges. The force grows with the size of each charge and falls off as the square of the distance between them: $F=kq_1q_2/r^2$ . Larger charges mean a bigger force; doubling the separation cuts the force to a quarter.

How do I recognize a Coulomb's Law problem?

It names two point charges (or charged spheres treated as points), gives both charge values and the separation $r$ , and asks for the force between them. The signature is 'two charges plus a distance, force as the answer.' Don't reach for it just because the word 'charge' appears — first confirm there are two interacting charges.

How is Coulomb's Law different from Electric Field?

Coulomb's Law is for the force between two specific charges that are both named. Electric Field uses a single source charge to describe what a test charge would feel at any point in space. If only one charge is given, you want the field ( $E=kQ/r^2$ ), not the two-charge force.

What is the most common mistake with Coulomb's Law?

Two slips. First, forgetting the inverse-square dependence — halving the distance quadruples the force, it does not just double it. Second, measuring $r$ between the surfaces of two charged spheres instead of centre to centre. Always use the centre-to-centre separation.

Section 12

Learning Path

← Before

Electric Charge Electric Field

Coulomb's Law

You are here

Electric Potential

Before this, students should be comfortable with Electric Charge and Electric Field. This page focuses on the recognition cue: Am I using a field or potential to explain how one object influences another across space? That cue connects earlier physical descriptions to later problem solving because students first choose the model, then choose the representation, equation, or explanation. After this, Electric Potential become easier to recognize.

Section 13