Faraday possessed extraordinary physical intuition despite lacking mathematical training, which gave him insight into how electromagnetic phenomena work without requiring formal equations.

Maxwell's third equation is Faraday's law: a changing magnetic field generates an electric field, which is the physical principle behind electromagnetic induction.

Maxwell's fourth equation states that electric currents generate magnetic fields, which was known from Ørsted's discovery that current-carrying wires deflect compass needles.

Maxwell's fourth equation (relating currents to magnetic fields) contains a logical inconsistency when applied to charging capacitors: in the gap between capacitor plates, no current flows, yet electric charge accumulates there creating an electric field; by the equation, a magnetic field should exist near the current in the wire but not across the capacitor gap, creating a discontinuity that is physically impossible.

Feynman claimed that Maxwell's equations would be remembered 10,000 years in the future as the defining achievement of the present era, even after everything else about our time is forgotten.

In the early 1800s, electricity and magnetism were understood as completely separate scientific domains with no apparent connection—electricity concerned charge phenomena like sparks and lightning, while magnetism concerned iron lodestones and compass needles.

In 1820, Danish physicist Hans Christian Ørsted discovered that electric current flowing through a wire produces a magnetic field, as evidenced by a compass needle deflecting when placed near a current-carrying wire during a lecture demonstration.

Michael Faraday was a self-taught scientist who began as a bookbinding apprentice, taught himself science by reading the books he was binding, and became one of history's greatest experimentalists despite having no formal mathematical training or calculus knowledge.

Faraday's discovery of electromagnetic induction is the physical principle underlying all electrical power generation: every power plant on Earth—coal, nuclear, wind, hydro—operates by moving a magnet past a coil of wire to induce electric current.

Faraday proposed that magnets act through invisible 'lines of force' or fields filling the space around them—a concept that was initially dismissed by mathematical physicists as poetic rather than rigorous, since Faraday could not defend it with mathematical equations.

James Clerk Maxwell was a Scottish mathematician of exceptional ability who took Faraday's field concept seriously and translated his physical intuitions into rigorous mathematical form, formulating equations that unified electricity and magnetism.

Maxwell's first equation states that electric charges are sources of electric fields: positive charges emit field lines pointing outward, negative charges emit field lines pointing inward, and electric field lines originate from and terminate at charges.

Maxwell's second equation states that magnetic fields have no sources (no magnetic charges exist): magnetic field lines never begin or end, always form closed loops, and consequently isolated magnetic poles cannot exist—breaking a magnet always produces new poles on each piece.

Maxwell resolved the inconsistency in his fourth equation by introducing the concept of 'displacement current'—a changing electric field acts on magnetic fields in the same way a conventional current does, even in the absence of moving charges, so that magnetic fields remain continuous across capacitor gaps.

Maxwell had no experimental evidence for the displacement current concept and introduced it solely because without it, his equations were mathematically inconsistent, demonstrating a bold willingness to modify fundamental laws based on mathematical consistency rather than empirical observation.

When the displacement current term is added to Maxwell's equations, the two 'bridge' equations (Faraday's law and the modified fourth equation) create a self-sustaining feedback loop: a changing magnetic field generates an electric field, which then changes and generates a new magnetic field, which changes again and generates another electric field, and so on indefinitely.

An electromagnetic wave can propagate through empty space without any material medium, requiring only the self-sustaining cycle of changing electric and magnetic fields generating each other as formalized in Maxwell's equations.

When Maxwell calculated the electromagnetic wave speed from electrical and magnetic constants, he obtained approximately 300,000 km/second, which is identical to the measured speed of light, providing the first theoretical evidence that light is an electromagnetic wave.

Physicists and scientists of Maxwell's era were convinced that all waves must propagate through a material medium, so they hypothesized the existence of the 'luminiferous ether'—an invisible substance filling all of space through which light waves traveled, despite having no empirical evidence for it.

Maxwell's equations logically entail that electromagnetic waves do not require any material medium—the electric and magnetic fields regenerate each other, holding each other up—meaning the luminiferous ether was unnecessary despite widespread scientific belief in its existence.