Electric Charge and Electric Force

SECTION 1

Historical Context & Motivation

The study of electric charge stretches back to the ancient Greeks, who observed that rubbed amber attracted lightweight objects such as straw and feathers. The Greek word for amber, ēlektron (ἤλεκτρον), eventually lent its name to the entire discipline of electricity. For nearly two millennia, however, these observations remained curiosities rather than elements of a systematic theory. It was not until the Scientific Revolution that natural philosophers began to distinguish electrical attraction from magnetism and to quantify the forces at play. The road from qualitative observation to a precise, mathematical law of electric force is one of the great narratives in the history of physics, culminating in Coulomb's law and the recognition of charge as a conserved, quantized property of matter.

~600 BCE

Thales of Miletus

Thales observed that amber, when rubbed with fur, attracted small objects—one of the earliest recorded investigations of electrostatic phenomena. This observation would remain qualitative for over two thousand years.

1600

William Gilbert's De Magnete

Gilbert distinguished electric attraction from magnetic attraction and coined the Latin term electricus. He catalogued many materials beyond amber that could be 'electrified' by friction.

1733

Du Fay's Two-Fluid Model

Charles François de Cisternay du Fay demonstrated the existence of two kinds of electricity—'vitreous' and 'resinous'—establishing that like charges repel and unlike charges attract.

1752

Benjamin Franklin's Unified Model

Franklin proposed a single-fluid model of electricity, introducing the conventions of positive and negative charge and demonstrating the electrical nature of lightning.

1785

Coulomb's Torsion-Balance Experiments

Charles-Augustin de Coulomb used an exquisitely sensitive torsion balance to show that the electric force between two point charges varies as the inverse square of the distance between them, placing electrostatics on firm quantitative ground.

With Coulomb's precise measurements, the central question became: what is charge, and how does it produce the force that holds atoms together, drives electric currents, and shapes every chemical reaction? This lesson explores the nature of electric charge itself—its conservation, quantization, and transfer mechanisms—and develops Coulomb's law as the quantitative framework for the electrostatic force between point charges.

SECTION 2

Core Principles & Definitions

Electric charge is one of the fundamental intrinsic properties of matter, analogous in importance to mass but differing in a crucial respect: charge comes in two varieties, conventionally labeled positive and negative. The interplay of these two signs gives rise to both attractive and repulsive forces, a richness that gravity—always attractive—lacks. Before developing the mathematical framework, we need to establish several foundational ideas that underpin all of electrostatics.

Two Kinds of Charge

All observed electric charges are either positive or negative. Like charges repel; unlike charges attract. Protons carry +e; electrons carry −e.

Conservation of Charge

The net electric charge in any isolated system is constant. Charge can be transferred between objects or created in particle-antiparticle pairs, but the algebraic sum never changes.

Quantization of Charge

Every observable charge is an integer multiple of the elementary charge e ≈ 1.602 × 10⁻¹⁹ C. Free quarks (with fractional charges) are confined and never observed in isolation.

Charging Mechanisms

Objects are charged by three primary mechanisms: friction (triboelectric transfer), conduction (direct contact), and induction (charge redistribution without contact).

Coulomb's Law

The electrostatic force between two point charges is proportional to the product of their charges and inversely proportional to the square of the distance between them. It acts along the line joining the charges.

✦ KEY TAKEAWAY

Think of electric charge as a kind of 'financial balance' that matter carries. Positive and negative charges are like credits and debits: you can shuffle them between accounts (objects), but the grand total across all accounts (the universe) never changes. Just as every monetary transaction is denominated in the smallest unit of currency, every charge transfer involves whole multiples of the elementary charge e. Coulomb's law then tells you the 'interest rate'—how strongly two charged accounts interact based on their balances and their separation.

SECTION 3

Visualizing Electric Force Between Charges

The diagram below illustrates how the Coulomb force operates between point charges. Two scenarios are shown side by side: the case of like charges (both positive), where the force is repulsive, and the case of unlike charges (one positive, one negative), where the force is attractive. The force vectors are drawn along the line connecting the charges, and their magnitudes scale with the product of the charges divided by the square of the separation distance.

Left panel: two positive charges experience a repulsive force directed away from one another. Right panel: a positive and a negative charge experience an attractive force directed toward one another. In both cases, the force magnitudes are equal (Newton's third law) and given by Coulomb's law.

Several features of the diagram merit attention. First, the force vectors on each charge are equal in magnitude but opposite in direction, consistent with Newton's third law—this holds even when q₁ ≠ q₂. Second, the force acts along the line connecting the two charges; there is no transverse component. Third, the direction of the force is determined by the signs of the charges: same signs yield repulsion, opposite signs yield attraction. The magnitude depends only on the absolute values |q₁| and |q₂|. These are the essential geometric and algebraic features that distinguish Coulomb's law from, say, the gravitational force, which is always attractive.

SECTION 4

Mathematical Framework

Coulomb's law gives the force between two stationary point charges. We present both the scalar (magnitude-only) and vector forms, along with the principle of superposition that allows us to handle systems of more than two charges.

COULOMB'S LAW — SCALAR FORM

F = k |q₁||q₂| / r²

F is the magnitude of the electrostatic force (N); k = 8.988 × 10⁹ N·m²/C² is Coulomb's constant; q₁ and q₂ are the charges (C); r is the center-to-center separation (m). The constant k is often written as 1/(4πε₀), where ε₀ ≈ 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space.

COULOMB'S LAW — VECTOR FORM

F⃗₁₂ = k q₁ q₂ / r² · r̂₁₂

Here F⃗₁₂ is the force on charge 2 due to charge 1, and r̂₁₂ is the unit vector pointing from charge 1 to charge 2. When q₁q₂ > 0 (like charges), F⃗₁₂ points away from q₁ (repulsion); when q₁q₂ < 0 (unlike charges), F⃗₁₂ points toward q₁ (attraction). The signs are embedded naturally in this vector expression.

SUPERPOSITION PRINCIPLE

F⃗_net = Σᵢ F⃗ᵢ = Σᵢ k q qᵢ / rᵢ² · r̂ᵢ

The net force on a charge q due to a collection of other charges is the vector sum of the individual Coulomb forces. Each pairwise force is computed independently, then all force vectors are added (component by component). This linearity is a deep and non-trivial property of electrostatics.

CHARGE QUANTIZATION

q = n e, n ∈ ℤ, e ≈ 1.602 × 10⁻¹⁹ C

Every macroscopic charge q is an integer multiple n of the elementary charge e. For an object with N_p protons and N_e electrons, the net charge is q = (N_p − N_e)e.

💡 Why 1/(4πε₀) instead of k?

Both notations are in common use and are entirely equivalent: k = 1/(4πε₀). The 4π factor arises naturally in Gauss's law, where the surface area of a sphere is 4πr². Using ε₀ simplifies many later results (e.g., capacitance, displacement field), while k keeps Coulomb's law compact. In this course you should be comfortable with both.

SECTION 5

Charging Mechanisms & the Inverse-Square Law

Understanding how objects acquire net charge is essential for applying Coulomb's law to real systems. There are three principal mechanisms: charging by friction (triboelectric effect), charging by conduction (direct contact with a charged object), and charging by induction (polarization of charge in response to a nearby charged object, followed by grounding). In friction charging, two dissimilar materials exchange electrons when rubbed together; the material with the stronger electron affinity strips electrons from the other, becoming negatively charged while the donor becomes positively charged. In conduction, free charges flow from a charged conductor to a neutral one upon contact until the system reaches electrostatic equilibrium. In induction, no material is transferred at all: a nearby charge polarizes a conductor, and selective grounding removes one sign of charge, leaving a net charge of the opposite sign after the ground connection is removed.

The inverse-square curve (yellow) shows how the Coulomb force between two fixed charges drops off rapidly with increasing separation r. At r = 2 (violet dot), the force is already only one-quarter of its value at r = 1 (red dot). The inset box summarizes the scaling behavior: halving the distance quadruples the force.

The inverse-square dependence is not unique to electrostatics; gravity follows the same functional form. This is not coincidental—in three spatial dimensions, both forces spread over the surface of a sphere whose area grows as 4πr², diluting the influence of a point source. The key quantitative difference is the enormous magnitude of the electric force relative to gravity. For two protons separated by 1 fm (10⁻¹⁵ m), the Coulomb repulsion exceeds the gravitational attraction by a factor of approximately 10³⁶. Electrostatics thus governs atomic and molecular structure, while gravity dominates at astronomical scales only because large bodies are nearly electrically neutral.

SECTION 6

Worked Example: Superposition of Forces

Consider three point charges arranged along the x-axis. Charge q₁ = +3.0 μC is at x = 0, charge q₂ = −5.0 μC is at x = 0.40 m, and charge q₃ = +2.0 μC is at x = 0.70 m. We wish to find the net electrostatic force on q₂ due to q₁ and q₃.

Net Force on q₂ via Superposition

Step 1 — Identify Given Valuesq₁ = +3.0 × 10⁻⁶ C at x₁ = 0 m; q₂ = −5.0 × 10⁻⁶ C at x₂ = 0.40 m; q₃ = +2.0 × 10⁻⁶ C at x₃ = 0.70 m. The separation between q₁ and q₂ is r₁₂ = 0.40 m, and between q₂ and q₃ is r₂₃ = 0.30 m. Coulomb's constant k = 8.99 × 10⁹ N·m²/C².

r₁₂ = 0.40 m, r₂₃ = 0.30 m

Step 2 — Force on q₂ Due to q₁ (F₂₁)F₂₁ = k|q₁||q₂|/r₁₂² = (8.99 × 10⁹)(3.0 × 10⁻⁶)(5.0 × 10⁻⁶)/(0.40)² = (8.99 × 10⁹)(15.0 × 10⁻¹²)/(0.16) = 0.1349/0.16 ≈ 0.843 N. Since q₁ is positive and q₂ is negative, q₂ is attracted toward q₁, which is in the −x direction.

F₂₁ = 0.843 N in the −x direction

Step 3 — Force on q₂ Due to q₃ (F₂₃)F₂₃ = k|q₂||q₃|/r₂₃² = (8.99 × 10⁹)(5.0 × 10⁻⁶)(2.0 × 10⁻⁶)/(0.30)² = (8.99 × 10⁹)(10.0 × 10⁻¹²)/(0.09) = 0.0899/0.09 ≈ 0.999 N. Since q₃ is positive and q₂ is negative, q₂ is attracted toward q₃, which is in the +x direction.

F₂₃ = 0.999 N in the +x direction

Step 4 — Apply Superposition (Vector Sum)Both forces lie along the x-axis. Taking +x as positive: F_net = F₂₃ − F₂₁ = +0.999 − 0.843 = +0.156 N. The net force is in the +x direction, toward q₃.

F_net ≈ 0.16 N in the +x direction

Step 5 — Physical InterpretationThe negative charge q₂ is pulled by both positive charges, but q₃ wins despite its smaller magnitude because it is closer (0.30 m vs. 0.40 m). The inverse-square dependence amplifies the proximity effect: the force from q₃ benefits from a factor of (0.40/0.30)² ≈ 1.78 relative to what equal magnitudes would give. This example illustrates how superposition and the 1/r² dependence together determine the net force.

SECTION 7

Coulomb Force vs. Gravitational Force

Students often find it instructive to compare the Coulomb force with Newton's law of universal gravitation, since both are central, inverse-square force laws. The structural similarity is striking—both depend on the product of two 'charges' (electric charge or mass) and both fall off as 1/r²—but their physical differences are profound.

Comparison of Coulomb's law and Newton's law of gravitation

Property	Coulomb Force (Electric)	Gravitational Force
Source quantity	Electric charge q (coulombs)	Mass m (kilograms)
Sign / Direction	Attractive or repulsive (two signs of charge)	Always attractive (mass is always positive)
Coupling constant	k ≈ 8.99 × 10⁹ N·m²/C²	G ≈ 6.674 × 10⁻¹¹ N·m²/kg²
Relative strength (proton-proton)	≈ 10³⁶ times stronger	1 (reference)
Dominant regime	Atomic & molecular scales; charged macroscopic objects	Planetary, stellar, and cosmological scales
Screening / Cancellation	Opposite charges neutralize; bulk matter is ~neutral	No cancellation—masses always add

✦ KEY TAKEAWAY

Despite their mathematical kinship, the Coulomb and gravitational forces occupy very different physical niches. The electric force is overwhelmingly stronger, but its two-sign nature means that bulk matter can achieve near-perfect neutrality, effectively 'hiding' the electric force at macroscopic scales. Gravity, though absurdly weak at the particle level, is unscreenable—every mass attracts every other mass—and therefore dominates cosmology. Engineering applications exploit this difference constantly: a battery (electric force) lifts electrons against a circuit's resistance with ease, while raising a satellite against Earth's gravity requires enormous expenditure of energy.

SECTION 8

Connection to Electric Fields & Advanced Theory

Coulomb's law, while complete for static point charges, is the foundation upon which the entire edifice of classical electromagnetism is built. The next conceptual step is to introduce the electric field E⃗, which reformulates the Coulomb force as a two-stage process: a source charge creates a field that permeates space, and a second charge immersed in that field experiences a force F⃗ = qE⃗. This field picture is more than a computational convenience—it becomes essential once charges are in motion, because changes in the field propagate at the speed of light rather than instantaneously, and the field itself carries energy and momentum.

From Coulomb's law to the field formulation

Concept	Coulomb's Law (This Lesson)	Field / Gauss's Law (Next Steps)
Interaction model	Direct action-at-a-distance between pairs of charges	Charge → field → force on test charge (mediated interaction)
Best suited for	Discrete point charges; few-body problems	Continuous charge distributions; high-symmetry geometries
Key equation	F = kq₁q₂/r²	∮ E⃗ · dA⃗ = Q_enc / ε₀
Handles moving charges?	Only in the static limit	Full Maxwell theory extends to time-varying fields
Energy perspective	Potential energy U = kq₁q₂/r	Energy density u = ½ε₀E² stored in the field

The transition from Coulomb's law to Gauss's law and eventually to the full set of Maxwell's equations represents one of the most beautiful unifications in all of physics. Gauss's law—which you will encounter in the next unit—restates Coulomb's law in integral form and exploits symmetry to solve problems that would be intractable by direct summation. The electric potential (voltage), capacitance, and the behavior of dielectrics all follow naturally from the concepts introduced here. Mastering the point-charge force law and the superposition principle therefore provides the conceptual and mathematical toolkit for the entire semester of electrostatics and beyond.

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

A glass rod is rubbed with silk, giving the rod a net positive charge. Explain, at the atomic level, what has happened to the rod and the silk. Is charge created in this process? How does the principle of conservation of charge apply?

PROBLEM 2 — BASIC CALCULATION

Two small spheres, each carrying a charge of +4.0 μC, are separated by a distance of 0.30 m in vacuum. Calculate the magnitude of the electrostatic force between them. Is the force attractive or repulsive?

PROBLEM 3 — INTERMEDIATE

Charge q₁ = +6.0 μC is located at the origin, and charge q₂ = −4.0 μC is at position (0.50 m, 0). A third charge q₃ = +2.0 μC is placed at position (0, 0.40 m). Find the magnitude and direction of the net electrostatic force on q₃.

PROBLEM 4 — APPLIED

In a simplified model of the hydrogen atom, the electron (charge −e) orbits the proton (charge +e) at a radius of 5.29 × 10⁻¹¹ m (the Bohr radius). Calculate the Coulomb force between the electron and proton, and compare it to the gravitational force between them. Use m_p = 1.673 × 10⁻²⁷ kg, m_e = 9.109 × 10⁻³¹ kg.

PROBLEM 5 — CRITICAL THINKING

Three identical positive charges +Q are placed at the vertices of an equilateral triangle of side length a. Show that the net force on each charge has magnitude F_net = k Q²√3 / a², and determine the direction of this force for one of the charges. Then consider: if the triangle is free to expand or contract, what would happen and why? Could a static arrangement of charges alone ever be in stable equilibrium?

SUMMARY

Lesson Summary

Electric charge is a fundamental, intrinsic property of matter that comes in two varieties: positive and negative. Charge is conserved—the net charge of an isolated system never changes—and quantized in integer multiples of the elementary charge e ≈ 1.602 × 10⁻¹⁹ C. Objects can be charged by friction, conduction, or induction, and in each case the total charge of the system is unchanged.

The force between two point charges is governed by Coulomb's law: F = k|q₁||q₂|/r², where k = 8.99 × 10⁹ N·m²/C². Like charges repel; unlike charges attract. For systems with more than two charges, the principle of superposition states that the net force on any charge is the vector sum of the individual pairwise Coulomb forces. Compared to gravity, the electric force is roughly 10³⁶ times stronger at the subatomic scale, but its two-sign nature allows bulk matter to achieve near-neutrality, relegating obvious electrical effects to situations where charge separation is deliberately maintained. These ideas—charge properties, Coulomb's law, and superposition—form the bedrock of electrostatics and the gateway to electric fields, Gauss's law, and Maxwell's equations.