Electromagnetism & Relativity
Course 2 · Chapter 1 · Electric Charge & Coulomb's Law
Classical Mechanics & Thermodynamics covered forces between masses. This course, the second half of the original "Classical Physics" split, opens with a different kind of force entirely — one that can both push and pull, and one whose 19th-century unification with magnetism directly set up the real crisis that led to relativity, covered later in this same course.
Electric Charge
Electric charge is a fundamental property of matter, coming in two kinds that either attract or repel. Charge is quantized in nature — every free charge is an exact whole-number multiple of the elementary charge, e:
e = 1.602176634 × 10&supminus;¹&sup9; C
Electric charge also obeys a real, exact conservation law: the total electric charge of an isolated system stays constant, regardless of whatever else changes within that system — charge can move from one place to another, or combine and separate in different ways, but the total amount never increases or decreases.
A Genuinely Arbitrary Naming Convention
Benjamin Franklin, working in the 1740s, coined the very term "charge" and proposed the labels "positive" and "negative" for the two kinds of electricity, based on a one-fluid theory of his own. His choice of which kind to call "positive" was, in his own era's own honest terms, an arbitrary convention — there was no physical reason at the time to prefer one label over the other.
⚠ Did Franklin "Guess Wrong"? Not Quite.
A popular version of this story claims Franklin's arbitrary guess turned out to be "backwards," since the electrons that actually carry current through a metal wire flow in the direction opposite to today's standard "conventional current" (defined as the direction positive charge would flow). This detail is real — conventional current genuinely does point opposite to real electron flow in a wire. But calling it a "wrong guess" overstates the case: a flow of positive charge in one direction has exactly the same measurable effect on a circuit as an equal flow of negative charge in the opposite direction. The convention still works correctly for every calculation; it just happens not to match which specific particle turned out to be the one actually moving, a fact nobody could have known in the 1740s, decades before the electron was even discovered.
Coulomb's Law
The force between two point charges is given by Coulomb's law:
F = kq1q2/r²
Here q1 and q2 are the two charges, r is the distance between them, and k is Coulomb's constant, 8.9875517862×10&sup9; N·m²·C&supminus;². Like charges repel; opposite charges attract — a genuine, structural difference from gravity, which is always attractive.
💡 A Real, Unpublished History Behind the Law
Charles-Augustin de Coulomb published this law in 1785, using a torsion balance — a bar suspended by a thin fiber, with charged metal-coated balls at each end — to measure the tiny electrostatic forces directly. But he was not the first to actually discover it: Joseph Priestley had proposed the same inverse-square relationship years earlier from his own experiments, without generalising or fully developing it, and Henry Cavendish — the same Cavendish credited in Classical Mechanics & Thermodynamics Chapter 6 with measuring the gravitational constant — had genuinely discovered the identical relationship in the early 1770s, over a decade before Coulomb, but simply never published it. Cavendish's later, famous 1797–98 gravity experiment used the very same torsion-balance technique Coulomb would go on to use for electric charge — a real, direct instrumental link between these two inverse-square laws, discovered by overlapping figures within a few decades of each other.
Worked Example: Force Between Two Charged Spheres
Two small spheres, carrying charges of +2 µC and +3 µC, are 0.5 m apart. What force do they exert on each other?
F = kq1q2/r²
F = (8.99×10&sup9;) × (2×10&supminus;&sup6;) × (3×10&supminus;&sup6;) / (0.5)²
F ≈ 0.216 N
Since both charges are positive, the force is repulsive — each sphere pushes the other away.
Worked Example: Proton and Electron in a Hydrogen Atom
A hydrogen atom's electron orbits its proton at roughly the Bohr radius, r ≈ 5.29×10&supminus;¹¹ m. Both particles carry a charge of magnitude e. What is the electric force between them?
F = ke²/r²
F = (8.99×10&sup9;) × (1.602×10&supminus;¹&sup9;)² / (5.29×10&supminus;¹¹)²
F ≈ 8.2×10&supminus;&sup8; N
🔗 Why Gravity Is Negligible Inside an Atom
Using Classical Mechanics & Thermodynamics Chapter 6's own gravitational force formula (F = Gm1m2/r²) on the same proton and electron (mp ≈ 1.673×10&supminus;²&sup7; kg, me ≈ 9.109×10&supminus;³¹ kg, same distance) gives a gravitational force of only about 3.6×10&supminus;&sup4;&sup7; N — the electric force between them is roughly 2×10³&sup9; times stronger than gravity. This is a real, genuinely enormous ratio, and it's exactly why gravity is utterly negligible at atomic scales, yet completely dominates at planetary and astronomical scales: individual particles carry a fixed, sizeable charge relative to their tiny mass, while large everyday objects and planets are almost perfectly charge-neutral overall, leaving gravity as the only significant force left standing once enough matter accumulates.
Coulomb's Law and Newton's Gravity: A Direct Structural Parallel
| Property | Newton's Gravity (Course 1, Ch.6) | Coulomb's Law (This Chapter) |
|---|
| Formula | F = Gm1m2/r² | F = kq1q2/r² |
| Distance dependence | Inverse-square | Inverse-square |
| Direction | Always attractive | Attractive or repulsive |
| Source property | Mass (always positive) | Charge (can be positive or negative) |
| Measured via | Torsion balance (Cavendish, 1797–98) | Torsion balance (Coulomb, 1785) |
Hands-On Exercises
Exercise 1
Two point charges, +5 uC and -4 uC, are separated by 0.3 m. Calculate the magnitude of the force between them, and state whether the force is attractive or repulsive.
→ Solution
Exercise 2
Two identical point charges, each of magnitude q, are 0.2 m apart and repel each other with a force of 9 N. Using Coulomb's law, find the magnitude of q.
→ Solution
Exercise 3
Explain, in your own words, why calling Benjamin Franklin's positive/negative naming convention a "wrong guess" is a real oversimplification - what genuinely IS backwards (relative to real electron flow), and what is NOT actually a problem for how circuits are calculated using the conventional-current convention today.
→ Solution
Quick Reference
- Elementary charge: e = 1.602176634×10&supminus;¹&sup9; C; all free charge is a whole-number multiple of it
- Conservation of charge: total charge in an isolated system never changes
- Franklin's positive/negative labels (1740s) were a genuinely arbitrary convention, not a provably "wrong" one
- Coulomb's law: F = kq1q2/r², k ≈ 8.99×10&sup9; N·m²·C&supminus;²
- Real history: Priestley (proposed, undeveloped) → Cavendish (discovered, unpublished, early 1770s) → Coulomb (published, 1785)
- Electric force between a proton and electron is roughly 2×10³&sup9; times stronger than gravity between them
Next chapter: Electric Fields & Potential — where the force Coulomb's law describes gets reframed as a field filling the space around a charge, and potential energy returns from Course 1's own Chapter 3 in an electrical form.