Electricity is a Circle

How all of electromagnetism falls out of geometry.

Out of the forces of the universe, electromagnetism is certainly the best understood. However, do not be fooled by this familiarity – it is still quite subtle.

The name electromagnetism clearly suggests that those two phenomena are linked, but if you were to ask the average person how, they would not be able to tell you. In truth, these two phenomena must be linked due to geometry. This is a beautiful truth and is sparsely appreciated, even by those who know physics.

Electric and magnetic forces were observed as far back as the Greeks. The phenomenon of static electricity was observed by rubbing amber (or in Greek, elektron, from which we derive the word electricity) which caused it to attract light objects like feathers. They also discovered magnetic iron ore, known as lodestones, which attracted other pieces of iron. Magnetic compasses were used as early as the 11th century by the Chinese for navigation.

Early Chinese compass, Song dynasty. (via: SSPL/Getty Images)

Up until around the 18th century, scientific understanding of electromagnetism remained almost purely phenomenological. The electric properties of various materials could be measured and documented - in particular, the affinities of various materials to gain attractive (electrostatic) properties when rubbed by another material, and the effects of humidity and heat. However, a quantitative, theoretical understanding was still lacking.

Arguably one of the first advances in this regard was by Charles-Augustin de Coulomb, who aimed to measure the strength of the electric force with distance. It had previously been assumed that the force decreased inversely with distance; however, Coulomb showed it was closer to the inverse distance cubed. Later, more delicate experiments would show that this force was actually inverse to the distance squared between charges q - a fact known as Coulomb’s law:

\(F=\frac{kq_1q_2}{r^2}\)

Later, Ørsted noticed that electrical current in a wire caused a magnetic compass to move. This inspired Ampere to investigate, and allowed him to establish Ampere's law explicitly relating electricity of magnetism. Such a suggestion of unification spurred the experimentalist Faraday to determine the extent of this relationship. Through this, amongst many other discoveries, he discovered the phenomenon of electromagnetic induction: moving magnets can create electric currents, a fact which allows for the generation of electricity that powers our modern world.

Maxwell took all of this experimental evidence and theory, and using his mathematical prowess, produced "A Treatise on Electricity and Magnetism" - a theory that mathematically unifies electricity and magnetism. This culminates in four equations, known as Maxwell's equations for electric (E) and magnetic (B) fields:

\( 1.\quad \text{div}(E)=\rho \quad \text{(Gauss' Law of Electricity)}\)

or, the electric field that flows, or diverges, from a point is sourced by electric charges,

\( 2.\quad \text{div}(B)=0\quad \text{(Gauss' Law of Magnetism)}\)

or, there is no way to create a net flow of magnetic field - north poles and south poles must always come in pairs,

\( 3.\quad \text{curl}(E)=-\frac{dB}{dt}\quad \text{(Faraday's Law)}\)

or, a changing magnetic field generates a curled electric field that induces a circular current,

\( 4.\quad \text{curl} (B)=j+\frac{dE}{dt}\quad \text{(Ampere-Maxwell Law)}\)

or, electric currents (j) and/or changing electric fields generate curled concentric magnetic fields around the wire. For example, below we see that switching on a current in the wire generates a magnetic field which deflects a compass needle away from north.