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Coulomb's Law

Master the equation that governs electric force between charged objects: how charge magnitude and separation distance determine attraction and repulsion.

High School Physics

What Is Electric Charge?

Electric charge is a fundamental property of matter. It comes in two types: positive (carried by protons) and negative (carried by electrons). Charge is measured in coulombs (C).

Like Charges
Two positive or two negative charges repel each other.
Unlike Charges
A positive and a negative charge attract each other.

Electric Force at a Distance

\(+q_1\) \(+q_2\) \(r\) \(F\) \(F\)

Charged objects exert forces on each other even when they are not touching. The force acts along the line connecting the two charges. Like charges repel: the forces point away from each other.

Why Coulomb's Law Matters

Coulomb's Law is the foundation of electrostatics. It tells us exactly how strong the electric force is between any two point charges. Every phenomenon from static cling to lightning to atomic structure relies on this law.

Chemistry
Explains why ions bond: opposite charges attract with calculable force.
Electronics
Governs how charges move through circuits and semiconductors.
Biology
Protein folding and nerve impulses depend on electrostatic interactions.

Coulomb's Law: The Equation

F
Electric force between the two charges, measured in newtons (N).
k
Coulomb's constant: \( k = 8.99 \times 10^9 \; \text{N} \cdot \text{m}^2 / \text{C}^2 \).
q
Charge magnitudes \( |q_1| \) and \( |q_2| \) in coulombs (C).
r
Distance between the centers of the two charges, in meters (m).

Visualizing the Inverse-Square Relationship

Distance \(r\) Force \(F\) Large F at small r Small F at large r

Coulomb's Law vs. Gravity

Coulomb's Law Newton's Gravity
Formula \( F = k\frac{|q_1||q_2|}{r^2} \) \( F = G\frac{m_1 m_2}{r^2} \)
Depends on Charge Mass
Direction Attract or repel Attract only
Relative strength Much stronger Much weaker
Distance dependence \( 1/r^2 \) \( 1/r^2 \)

Attraction vs. Repulsion

+ − Attract + + Repel

The sign of the charges determines direction. Coulomb's Law gives only the magnitude of the force. You determine direction by checking whether the charges are like (repel) or unlike (attract).

How to Apply Coulomb's Law

1
Identify the two charges \( q_1 \) and \( q_2 \) and record their magnitudes in coulombs.
2
Measure the distance \( r \) between the centers of the charges in meters.
3
Substitute into the formula: \( F = k \frac{|q_1||q_2|}{r^2} \), where \( k = 8.99 \times 10^9 \).
4
Determine the direction: like charges repel, unlike charges attract.
5
Check units: charges in C, distance in m, force result in N.

Example 1: Basic Calculation

\(+3\,\mu\text{C}\) \(q_1\) \(-5\,\mu\text{C}\) \(q_2\) \(r = 0.2\,\text{m}\)

Given: \( q_1 = +3 \;\mu\text{C} = 3 \times 10^{-6} \;\text{C} \), \( q_2 = -5 \;\mu\text{C} \), so \( |q_2| = 5 \times 10^{-6} \;\text{C} \), \( r = 0.2 \;\text{m} \).

Example 2: Doubling the Distance

Question: Using the same charges from Example 1, what happens to the force if we double the distance to \( r = 0.4 \;\text{m} \)?

Example 3: Finding the Distance

Problem: Two charges \( q_1 = 4 \;\mu\text{C} \) and \( q_2 = 6 \;\mu\text{C} \) repel with a force of \( F = 5.4 \;\text{N} \). Find the distance \( r \).

Multiple Charges: Superposition

\(q_1\) \(q_2\) \(q_3\) \(F_{12}\) \(F_{32}\) \( \vec{F}_{\text{net on } q_2} = \vec{F}_{12} + \vec{F}_{32} \)

When more than two charges are present, the net force on any charge is the vector sum of the individual forces from every other charge. Calculate each pair separately using Coulomb's Law, then add them.

Units and Conversions

Prefix Symbol Factor Example
Microcoulomb \( \mu\text{C} \) \( 10^{-6} \;\text{C} \) \( 3 \;\mu\text{C} = 3 \times 10^{-6} \;\text{C} \)
Nanocoulomb \( \text{nC} \) \( 10^{-9} \;\text{C} \) \( 50 \;\text{nC} = 5 \times 10^{-8} \;\text{C} \)
Picocoulomb \( \text{pC} \) \( 10^{-12} \;\text{C} \) \( 200 \;\text{pC} = 2 \times 10^{-10} \;\text{C} \)
Centimeters \( \text{cm} \) \( 10^{-2} \;\text{m} \) \( 30 \;\text{cm} = 0.3 \;\text{m} \)

Common Mistakes

✕ Forgetting Unit Conversions
Using \( \mu\text{C} \) or cm directly in the formula instead of converting to C and m first.
✕ Squaring the Wrong Value
Remember: it is the distance \( r \) that is squared, not the charge \( q \).
✕ Ignoring Direction
Reporting only magnitude without stating whether the force is attractive or repulsive.

Key Takeaways

The Formula
\( F = k \frac{|q_1||q_2|}{r^2} \) with \( k = 8.99 \times 10^9 \)
Inverse-Square
Double the distance: force drops to \( \frac{1}{4} \). Triple: \( \frac{1}{9} \).
Direction Rule
Like charges repel, unlike charges attract. Formula gives magnitude only.
Units Matter
Convert to coulombs and meters before plugging in. Check dimensions.

Question 1 of 5

Two charges of \( +2 \;\mu\text{C} \) and \( +8 \;\mu\text{C} \) are separated by \( 0.3 \;\text{m} \). What is the magnitude of the electric force between them?

Question 2 of 5

If the distance between two charges is tripled, what happens to the electric force?

Question 3 of 5

A charge of \( +4 \;\mu\text{C} \) and a charge of \( -4 \;\mu\text{C} \) are separated by \( 0.1 \;\text{m} \). The force between them is:

Question 4 of 5

In Coulomb's Law, the constant \( k \) has the value:

Question 5 of 5

Two identical charges experience a force of \( 10 \;\text{N} \) when \( 0.5 \;\text{m} \) apart. If one charge is doubled and the distance is halved, the new force is: