The previous part showed that electric and magnetic fields store energy and resist change. This part asks a deeper question: what are they, really? The answer leads through Maxwell's equations to light, and then to Einstein's discovery that the two fields are not separate things at all.
The symmetry Maxwell completed
Faraday had found one direction of the coupling experimentally: a changing magnetic field creates an electric field. Spin a coil in a magnetic field and voltage appears. Proven, measured, used in every generator ever built.
Maxwell asked whether the reverse was also true: does a changing electric field create a magnetic field? There was no experimental evidence. But without this term, Maxwell noticed his equations were mathematically inconsistent: they violated conservation of charge in certain situations. He added the missing piece on theoretical grounds alone.
The result was a perfect symmetry:
Faraday's law
Flux is the amount of field passing through a surface: field strength times area times the cosine of the angle between the field and the surface normal. Field straight through: maximum flux. Field parallel to the surface: zero flux.
Faraday's law says: a changing magnetic flux through a loop induces a voltage around that loop. Not the field itself, but the rate of change of flux. A steady magnetic field through a loop does nothing. Start changing it and a voltage appears.
The minus sign is Lenz's law: the induced voltage always opposes the change that caused it. Increase the flux and the induced current fights the increase. This is the inductor's defining behaviour; the coil is Faraday's law made physical. It is also how transformers work: alternating current in one coil creates a changing magnetic field, that field threads through the second coil, changing flux induces voltage there. No electrical connection needed.
Ampere's law and Maxwell's addition
Ampere's law in its original form: the magnetic field around a closed loop equals the total current passing through the surface that loop encloses.
For a straight wire this gives the field at distance r directly: B = μ₀I/(2πr). Closer to the wire, stronger field. More current, stronger field.
Maxwell found a contradiction in this law at a capacitor. Current flows in the wire toward the plate; Ampere's law gives a magnetic field around that wire. But draw the surface through the gap between the plates instead of through the wire: no current crosses that surface. The same loop, the same boundary, should give the same magnetic field, but the original law gives zero.
The resolution: as the capacitor charges, the electric field in the gap is growing. Maxwell showed that a changing electric flux acts identically to a real current for the purposes of creating a magnetic field. He called it displacement current and added it to Ampere's law:
For the surface through the wire: the first term gives the answer, the second is zero. For the surface through the gap: the first term is zero, the second gives the same answer, because the rate of change of electric flux in the gap exactly equals the current in the wire. Both surfaces, same loop, same result. The contradiction is resolved, and the symmetry is complete.
The wave
With the symmetry complete, something new becomes possible. A changing E creates B. That B is also changing, because the E that produced it is oscillating. That changing B creates E a little further out. That E is changing, so it creates B further still. Each field generates the next. The disturbance propagates outward through empty space, carrying itself, with no charge or wire required.
This is an electromagnetic wave. The two fields oscillate at the same frequency, in phase, each perpendicular to the other and both perpendicular to the direction of travel. E points in one direction, B points at right angles to it, the wave travels at right angles to both.
The speed at which this leapfrog propagates is set by how strongly each field responds to a change in the other, which is exactly what ε₀ and μ₀ measure. Put them together:
Maxwell calculated this from constants measured with torsion balances and current-carrying wires, with no reference to light. He got 3 × 10⁸ m/s. The speed of light had been measured independently. They matched. Maxwell realised light is an electromagnetic wave, not similar to one but identical.
The spectrum
A charge oscillating at any frequency produces a wave at that frequency. All frequencies are the same phenomenon:
| Name | Frequency | Wavelength |
|---|---|---|
| Radio | kHz to GHz | km to cm |
| Microwave | GHz range | cm |
| Infrared | THz range | μm |
| Visible light | 430–770 THz | 700–400 nm |
| Ultraviolet | higher | shorter |
| X-ray | much higher | nm to pm |
| Gamma | highest | pm and below |
One field, one mechanism, the entire spectrum. The only difference is frequency.
One field, two faces
A stationary charge creates only an electric field. Move past that same charge and it is no longer stationary from your point of view; it is a moving charge, which means a current, which means a magnetic field. Same charge, different observer, different fields measured.
Einstein showed in 1905 that this is not a coincidence or an approximation. Electric and magnetic fields are two aspects of one underlying thing: the electromagnetic field. Which aspect you observe depends on how you are moving relative to the source. What one observer measures as purely electric, another measures as a mixture of electric and magnetic. They are not separate phenomena that happen to interact. They are one field, seen from different angles in spacetime.
This is why ε₀ and μ₀ are related through c; they are not two independent constants but two faces of one constant, tied by c² = 1/(ε₀μ₀). Maxwell's equations already encoded this unity. Einstein made it explicit.
Light is what the electromagnetic field looks like when its two faces sustain each other through empty space, propagating at c with no medium and no source required.
Where this comes from
- Maxwell's equations and electromagnetic waves: Griffiths, Introduction to Electrodynamics, ch. 8 to 9.
- Historical development and physical intuition: Feynman, Leighton, Sands, The Feynman Lectures on Physics, vol. 2, ch. 18 to 20.
- Special relativity and electromagnetism: Purcell and Morin, Electricity and Magnetism, ch. 5 to 6.
Check yourself
1. Faraday's law says a changing magnetic flux induces a voltage. What happens if the flux through a coil is large but constant?
No voltage is induced. Faraday's law depends on the rate of change of flux, not the flux itself. A steady field, however strong, induces nothing.
2. A capacitor is being charged. No current crosses the gap between the plates, yet the magnetic field around the gap is nonzero. What creates it?
The growing electric field in the gap is Maxwell's displacement current. A changing electric flux acts identically to a real current in Ampere's law, producing the same magnetic field as if current were flowing through the gap.
3. A radio antenna broadcasts at 100 MHz. An engineer checks the antenna and finds it also radiates weakly at 200 MHz and 300 MHz. What are these, and why do they occur?
Harmonics. Any non-sinusoidal oscillation contains integer multiples of the fundamental frequency. Non-linear components in the transmitter distort the sine wave, generating harmonics. They are weaker than the fundamental but real, and must be filtered to avoid interfering with other frequency bands.
4. Observer A stands still beside a current-carrying wire and measures a magnetic field. Observer B moves alongside the electrons at the same speed and measures an electric field. Which observer is correct?
Both are correct. Electric and magnetic fields are aspects of one electromagnetic field; which components appear depends on the observer's velocity relative to the source. Neither measurement is more real than the other.