Coulomb's Law Equation, Derivation & Rearrangements

Core idea

Overview

Coulomb's Law quantifies the force of attraction or repulsion between two electrically charged particles. It states that the magnitude of the electrostatic force (F) between two point charges is directly proportional to the product of the magnitudes of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them. The constant of proportionality, k, is Coulomb's constant, which depends on the medium. This fundamental law is crucial for understanding electric fields, potential, and the behavior of charged particles in various physical systems.

When to use: Apply this formula when you need to determine the electrostatic force between two point charges, given their magnitudes and the distance separating them. It's also used to find an unknown charge or distance if the force and other variables are known. Ensure charges are in Coulombs and distance in meters.

Why it matters: Coulomb's Law is foundational to electromagnetism, explaining phenomena from atomic structure to the operation of electronic devices. It's essential for understanding how charges interact, forming the basis for electric circuits, capacitors, and the behavior of matter at a microscopic level. Its principles are applied in fields like materials science, particle physics, and electrical engineering.

Symbols

Variables

k = Coulomb's Constant, q_1 = Charge 1, q_2 = Charge 2, r = Distance between charges, F = Electrostatic Force

k

Coulomb's Constant

N m^{2} / C^{2}

q_{1}

Charge 1

C

q_{2}

Charge 2

C

r

Distance between charges

m

F

Electrostatic Force

N

Walkthrough

Derivation

Formula: Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges.

The charges are point charges (or spherically symmetric charge distributions).
The medium between the charges is uniform and isotropic (often vacuum or air).
The charges are stationary (electrostatic conditions).

1

Experimental Observation:

Charles-Augustin de Coulomb's experiments showed that the force between two charges is directly proportional to the product of their magnitudes.

F \propto ∣ q_{1} q_{2} ∣

2

Inverse Square Law:

He also found that the force is inversely proportional to the square of the distance between the charges.

F \propto \frac{1}{r ^{2}}

3

Combining Proportionalities:

Combining these observations gives the overall proportionality.

F \propto \frac{∣ q _{1} q _{2} ∣}{r ^{2}}

4

Introducing Coulomb's Constant:

To turn the proportionality into an equation, a constant of proportionality, k (Coulomb's constant), is introduced. This constant depends on the medium and the system of units used.

F = k \frac{∣ q _{1} q _{2} ∣}{r ^{2}}

Result

F = k \frac{∣ q _{1} q _{2} ∣}{r ^{2}}

Source: AQA A-level Physics — Electric Fields (7408/7407)

Free formulas

Rearrangements

Solve for $k$

Coulomb's Law: Make k the subject

k = \frac{F r ^{2}}{∣ q _{1} q _{2} ∣}

To make $k$ (Coulomb's constant) the subject of Coulomb's Law, multiply both sides by $r^{2}$ and then divide by $∣ q_{1} q_{2} ∣$ .

Difficulty: 2/5

Solve for $q_{1}$

Coulomb's Law: Make q1 the subject

∣ q_{1} ∣ = \frac{F r ^{2}}{k ∣ q _{2} ∣}

To make $∣ q_{1} ∣$ (magnitude of charge 1) the subject of Coulomb's Law, multiply both sides by $r^{2}$ , then divide by $k$ and $∣ q_{2} ∣$ .

Difficulty: 3/5

Solve for $q_{2}$

Coulomb's Law: Make q2 the subject

∣ q_{2} ∣ = \frac{F r ^{2}}{k ∣ q _{1} ∣}

To make $∣ q_{2} ∣$ (magnitude of charge 2) the subject of Coulomb's Law, multiply both sides by $r^{2}$ , then divide by $k$ and $∣ q_{1} ∣$ .

Difficulty: 3/5

Solve for $r$

Coulomb's Law: Make r the subject

r = \frac{k ∣ q _{1} q _{2} ∣}{F}

To make $r$ (distance) the subject of Coulomb's Law, multiply both sides by $r^{2}$ , divide by $F$ , and then take the square root.

Difficulty: 3/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Imagine two tiny, charged spheres either pulling towards each other or pushing apart along the straight line connecting their centers, with the strength of this interaction becoming much weaker as they move further

q_{1}

The magnitude of the electric charge of the first point particle.

The 'amount' of electric charge on the first particle. More charge leads to a stronger force.

q_{2}

The magnitude of the electric charge of the second point particle.

The 'amount' of electric charge on the second particle. More charge leads to a stronger force.

r

The distance separating the centers of the two point charges.

How far apart the charges are. The force rapidly decreases as this distance increases.

k

Coulomb's constant, a proportionality constant that depends on the medium and the system of units.

A fundamental constant that scales the force, reflecting the strength of the electromagnetic interaction in a vacuum (or specific medium).

Signs and relationships

|q_1 q_2|: The absolute value ensures that the calculated force F is always a positive magnitude. The direction of the force (attraction or repulsion)
r^2 (in the denominator): This inverse square dependence means the electrostatic force weakens very rapidly as the distance between the charges increases. Doubling the distance reduces the force to one-fourth of its original value.

Free study cues

Insight

Canonical usage

This equation is normally used to calculate electrostatic force, with specific unit systems dictating the value and units of Coulomb's constant (k) and the units of force, charge, and distance.

Common confusion

A common mistake is mixing units from different systems (e.g., using Coulombs for charge with k=1 from CGS) or using an incorrect value/unit for Coulomb's constant (k).

Unit systems

$F$ Newtons (N) in SI; dynes (dyn) in CGS · Force is a vector quantity, but Coulomb's Law calculates its magnitude.

$q 1, q 2$ Coulombs (C) in SI; statcoulombs (esu) in CGS · The absolute value ensures the magnitude of the force is calculated, regardless of charge signs.

$r$ meters (m) in SI; centimeters (cm) in CGS · Distance between the centers of the two point charges.

Ballpark figures

Quantity:

One free problem

Practice Problem

Two point charges, $q_{1} = 2.0 μ C$ and $q_{2} = 3.0 μ C$ , are separated by a distance of $5.0$ cm. Calculate the magnitude of the electrostatic force between them in a vacuum. Use Coulomb's constant $k = 8.987 \times 1 0^{9} N m^{2} / C^{2}$ . Round to two decimal places.

Charge 10.000002 C

Charge 20.000003 C

Distance between charges0.05 m

Coulomb's Constant8987000000 N m^2/C^2

Solve for: $F$

Hint: Remember to convert microcoulombs to coulombs and centimeters to meters.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating the force between electrons in an atom or the force between charged particles in a particle accelerator.

Study smarter

Tips

Remember that force is a vector quantity; Coulomb's Law gives the magnitude. The direction is along the line connecting the charges (repulsive for like charges, attractive for opposite charges).
The constant 'k' (Coulomb's constant) is approximately $8.987 \times 1 0^{9} N m^{2} / C^{2}$ in a vacuum.
Ensure all units are in SI: Coulombs (C) for charge, meters (m) for distance, and Newtons (N) for force.
The absolute value in the formula means the force magnitude is always positive, regardless of charge signs.

Avoid these traps

Common Mistakes

Forgetting to square the distance 'r'.
Not using the absolute value for charges, leading to negative force magnitudes (which is incorrect for magnitude).
Mixing units (e.g., cm for distance instead of m, microcoulombs instead of Coulombs without conversion).
Confusing Coulomb's Law with Newton's Law of Universal Gravitation (similar inverse square relationship but different forces and constants).

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Coulomb's Law describes the electrostatic force between two point charges.

Apply this formula when you need to determine the electrostatic force between two point charges, given their magnitudes and the distance separating them. It's also used to find an unknown charge or distance if the force and other variables are known. Ensure charges are in Coulombs and distance in meters.

Coulomb's Law is foundational to electromagnetism, explaining phenomena from atomic structure to the operation of electronic devices. It's essential for understanding how charges interact, forming the basis for electric circuits, capacitors, and the behavior of matter at a microscopic level. Its principles are applied in fields like materials science, particle physics, and electrical engineering.

Forgetting to square the distance 'r'. Not using the absolute value for charges, leading to negative force magnitudes (which is incorrect for magnitude). Mixing units (e.g., cm for distance instead of m, microcoulombs instead of Coulombs without conversion). Confusing Coulomb's Law with Newton's Law of Universal Gravitation (similar inverse square relationship but different forces and constants).

Calculating the force between electrons in an atom or the force between charged particles in a particle accelerator.

Remember that force is a vector quantity; Coulomb's Law gives the magnitude. The direction is along the line connecting the charges (repulsive for like charges, attractive for opposite charges). The constant 'k' (Coulomb's constant) is approximately $8.987 \times 10^9 \text{ N m}^2/\text{C}^2$ in a vacuum. Ensure all units are in SI: Coulombs (C) for charge, meters (m) for distance, and Newtons (N) for force. The absolute value in the formula means the force magnitude is always positive, regardless of charge signs.

References

Sources

Halliday, Resnick, Walker, Fundamentals of Physics
Griffiths, Introduction to Electrodynamics
Wikipedia: Coulomb's law
Encyclopædia Britannica: Coulomb's law
NIST CODATA (for k in SI)
Fundamentals of Physics by Halliday, Resnick, and Walker
Introduction to Electrodynamics by David J. Griffiths
Coulomb's law (Wikipedia article title)

Coulomb's Law

Overview

Variables

Derivation

Experimental Observation:

Inverse Square Law:

Combining Proportionalities:

Introducing Coulomb's Constant:

Rearrangements

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Electric Field Strength

Electric Potential Energy

Capacitance

Frequently Asked Questions

Sources