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Topic 1 of 15

Electric Charge & Coulomb's Law

Electric charge is one of the fundamental properties of matter. The universe contains two types of charge — positive and negative — and the force between them is described by Coulomb's law, an inverse-square law remarkably similar in form to gravity but enormously stronger. Understanding charge, its conservation, and the forces it produces is the foundation of all electromagnetism.

Key Concepts

Electric Charge

A fundamental property of matter. Two types: positive (protons) and negative (electrons). Like charges repel; unlike charges attract. Charge is quantized — it comes in integer multiples of the elementary charge

e = 1.602\times10^{-19}

C. Charge is conserved: the total charge of an isolated system never changes.

Conductors & Insulators

Conductors (metals) have free electrons that move easily throughout the material. Insulators (rubber, glass) have electrons tightly bound to atoms. Semiconductors (silicon) fall in between and are controlled by doping.

Charging by Induction

A conductor can be charged without contact. A nearby charged object polarizes the conductor (pushing or pulling free electrons), and if the conductor is then grounded momentarily, it retains a net charge when the ground is removed.

Coulomb's Law

The electrostatic force between two point charges

q_1

and

q_2

separated by distance

r

has magnitude

F = k|q_1||q_2|/r^2

, where

k = 8.99\times10^9

N·m²/C². The force is repulsive for like charges and attractive for unlike charges, directed along the line joining them.

Superposition Principle

The net force on a charge due to multiple other charges is the vector sum of the individual Coulomb forces. Forces from different pairs are independent and add as vectors.

Key Equations

Coulomb's Law

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

Magnitude of the electrostatic force between point charges q₁ and q₂ separated by distance r.

Coulomb's constant

k = \frac{1}{4\pi\varepsilon_0} = 8.99\times10^9 \text{ N·m}^2\text{/C}^2

ε₀ = 8.85×10⁻¹² C²/(N·m²) is the permittivity of free space.

Elementary charge

e = 1.602\times10^{-19} \text{ C}

Magnitude of the charge of one electron (negative) or one proton (positive). All observable charges are multiples of e.

Worked Example

Force Between Two Point Charges

Problem

Charge $q_1 = +3\,\mu\text{C}$ is at the origin and $q_2 = -5\,\mu\text{C}$ is at $x = 0.20$ m. Find the magnitude and direction of the force on $q_1$ .

Solution

Apply Coulomb's Law for the magnitude:

F = k\frac{|q_1||q_2|}{r^2} = (8.99\times10^9)\frac{(3\times10^{-6})(5\times10^{-6})}{(0.20)^2}

F = (8.99\times10^9)\frac{15\times10^{-12}}{0.04} = \frac{134.85\times10^{-3}}{0.04} = 3.37 \text{ N}

The charges are opposite, so the force is attractive: $q_1$ is pulled toward $q_2$ , i.e., in the $+x$ direction.

Answer F = 3.37 N, directed toward q₂ (in the +x direction).

Practice

Exercises

7 problems

1 of 7

Two positive point charges $q_1 = +4\,\mu\text{C}$ and $q_2 = +6\,\mu\text{C}$ are separated by $r = 0.30$ m. What is the magnitude of the electrostatic force (in N) between them?

Your answer

2 of 7

A metal sphere has $5.0\times10^{12}$ excess electrons placed on it. What is the magnitude of the net charge (in nC)?

Your answer

3 of 7

Two identical positive charges are $r = 0.50$ m apart and repel each other with force $F = 0.090$ N. What is the magnitude of each charge (in $\mu$ C)?

Your answer

μC

4 of 7

Charge $q_1 = +2\,\mu\text{C}$ is at the origin and $q_2 = -3\,\mu\text{C}$ is at $x = 0.40$ m. What is the magnitude of the force (in N) on $q_1$ ?

Your answer

5 of 7

Charges $q_1 = +8\,\mu\text{C}$ (at $x = 0$ ) and $q_2 = +2\,\mu\text{C}$ (at $x = 0.30$ m) are fixed. At what position $x$ (in m) between them does a positive test charge experience zero net force?

Your answer

6 of 7

A charge $q_1 = +5\,\mu\text{C}$ feels a repulsive force of $0.45$ N from another charge at $r = 0.20$ m. What is the magnitude of that other charge (in $\mu$ C)?

Your answer

μC

7 of 7

Two charges $q_1 = q_2 = 2.0\,\mu\text{C}$ repel with force $F = 36$ N. What is their separation (in cm)?

Your answer

Key Takeaways

Charge is quantized (integer multiples of $e = 1.602\times10^{-19}$ C) and conserved in all interactions.
Coulomb's law $F = k|q_1||q_2|/r^2$ is an inverse-square law — doubling the distance reduces the force by a factor of 4.
The electrostatic force is $\sim 10^{36}$ times stronger than gravity between two protons — gravity only dominates at cosmic scales because large objects are nearly charge-neutral.
Superposition: forces from multiple charges add as vectors — set up components and sum separately.
For equilibrium problems, set the net force (or its components) equal to zero and solve.