Why electrodynamics: from static fields to light

PHYS-381 built electrostatics and magnetostatics on four field equations, two for the electric field and two for the magnetic field, with the time derivatives switched off. In that static world the two fields are separate subjects: charges at rest make electric fields, steady currents make magnetic fields, and neither cares about the other. This course switches time back on. Once fields are allowed to change, a changing magnetic field produces an electric field, a changing electric field produces a magnetic field, and the two together can travel through empty space as light. Electrodynamics is the theory of that coupling. It explains generators and transformers, the energy that flows through a power cable, the colour of the sky, the way a microwave oven heats food, how antennas send and receive signals, and why the speed of light is the same for every observer. This first lesson sets out where the static theory breaks down, previews the full set of Maxwell's equations, and explains when the old static results are still good enough.

What the static theory leaves out

The static equations of PHYS-381 were

Two experimental facts show that they cannot be the whole story. The first is electromagnetic induction. In 1831 Michael Faraday found that moving a magnet near a coil, or switching on a current in a neighbouring coil, drives a current through the coil even though no battery is connected. A current needs a force on the charges, and since the coil is at rest that force must be electric. The electric field that drives it circulates around the coil, so its curl cannot be zero. The static law \(\nabla\times\mathbf{E} = \mathbf{0}\) must acquire a term that involves the changing magnetic field.

The second fact is subtler, and it was found on paper rather than in the laboratory. Take the divergence of the static Ampère law. Since the divergence of any curl vanishes, \(\nabla\cdot(\nabla\times\mathbf{B}) = 0\), the law demands \(\nabla\cdot\mathbf{J} = 0\): current can never pile up anywhere. That is true for steady currents, but it is false for a capacitor that is charging, where current flows into one plate and charge accumulates there. The equation of continuity, which expresses conservation of charge, says instead

So the static Ampère law contradicts conservation of charge whenever charge densities change. James Clerk Maxwell repaired it in 1861 and 1862 by adding a term, the displacement current, that he first justified with a mechanical model of the ether and later showed was needed for consistency alone. Lesson 3 derives it. With both repairs in place the equations predict waves travelling at \(1/\sqrt{\varepsilon_0\mu_0}\), a speed Maxwell computed from electrical measurements and found equal to the measured speed of light. In his 1865 paper A Dynamical Theory of the Electromagnetic Field he concluded that light is an electromagnetic disturbance.

Maxwell's equations

The full equations, in vacuum with sources \(\rho\) and \(\mathbf{J}\), are

and together with the Lorentz force \(\mathbf{F} = q(\mathbf{E} + \mathbf{v}\times\mathbf{B})\) they contain all of classical electromagnetism. Maxwell's own presentation used twenty equations in twenty unknowns. The compact vector form was introduced by Oliver Heaviside and, independently, by Heinrich Hertz in the 1880s, and it is the form every physicist now learns. The two divergence equations are unchanged from statics. The curl equations now couple the fields: each field's circulation is fed by the rate of change of the other. That coupling is the whole of the new physics.

Read each equation as a statement about a small region of space. Gauss's law says electric field lines start on positive charge and end on negative charge. The second equation says magnetic field lines never start or end. Faraday's law says a magnetic field that is changing in time is wrapped by a circulating electric field, with the sign fixed so that induced currents oppose the change; this is Lenz's rule. The Ampère–Maxwell law says magnetic field circulates around both real currents and changing electric fields. The equations are linear in the fields and sources, so superposition still holds.

When is the static picture good enough?

Most of electrical engineering at low frequency uses static ideas: Kirchhoff's laws, capacitance, inductance and resistance treated as fixed numbers. That works because two different effects are small. The first is the size of the displacement current compared with the conduction current. In a material with conductivity \(\sigma\) and permittivity \(\varepsilon\), an oscillating field of angular frequency \(\omega\) drives a conduction current density \(\sigma E\) and a displacement current density of magnitude \(\omega\varepsilon E\). Their ratio \(\omega\varepsilon/\sigma\) tells you whether the material behaves as a conductor or as a dielectric. The second effect is retardation: fields take time to travel, and if a circuit is a sizeable fraction of a wavelength across, different parts of it see the source at different moments, and radiation becomes important.

The program below checks both. It first confirms that \(1/\sqrt{\varepsilon_0\mu_0}\) is the speed of light. It then tabulates \(\omega\varepsilon/\sigma\) for copper, sea water and dry soil at three frequencies, compares two circuits with the wavelength at their operating frequency, and finally lists how long field changes take to cross some familiar distances.

import math
eps0 = 8.8541878188e-12; mu0 = 1.25663706212e-6
c = 1 / math.sqrt(eps0 * mu0)
print(f"c = 1/sqrt(eps0 mu0) = {c:.6e} m/s")

# Ratio of displacement current to conduction current, omega*eps/sigma,
# for a material with conductivity sigma and relative permittivity er.
materials = [("copper", 5.96e7, 1.0), ("sea water", 4.0, 80.0), ("dry soil", 1e-4, 5.0)]
freqs = [60.0, 1e6, 1e10]
print("material      60 Hz        1 MHz        10 GHz")
for name, sigma, er in materials:
    row = [2 * math.pi * f * er * eps0 / sigma for f in freqs]
    print(f"{name:11s}" + "".join(f"  {r:11.2e}" for r in row))

# Quasistatic test: is the circuit much smaller than a wavelength?
for label, f, size in [("power grid, 60 Hz, 1000 km line", 60.0, 1.0e6),
                       ("phone circuit board, 2.4 GHz, 5 cm", 2.4e9, 0.05)]:
    lam = c / f
    print(f"{label}: wavelength {lam:.3g} m, size/wavelength = {size / lam:.3g}")

# How long changes in the field take to arrive
for label, d in [("Earth to Moon", 3.844e8), ("Sun to Earth", 1.496e11), ("across a 30 cm circuit board", 0.30)]:
    print(f"light travel time, {label}: {d / c:.3g} s")

c = 1/sqrt(eps0 mu0) = 2.997925e+08 m/s
material      60 Hz        1 MHz        10 GHz
copper          5.60e-17     9.33e-13     9.33e-09
sea water       6.68e-08     1.11e-03     1.11e+01
dry soil        1.67e-04     2.78e+00     2.78e+04
power grid, 60 Hz, 1000 km line: wavelength 5e+06 m, size/wavelength = 0.2
phone circuit board, 2.4 GHz, 5 cm: wavelength 0.125 m, size/wavelength = 0.4
light travel time, Earth to Moon: 1.28 s
light travel time, Sun to Earth: 499 s
light travel time, across a 30 cm circuit board: 1e-09 s

Copper is a conductor by a vast margin at every frequency in the table; even at 10 GHz the displacement current inside the metal is about one hundred-millionth of the conduction current. Sea water is a good conductor at power and radio frequencies but behaves more like a lossy dielectric at microwave frequencies, where the ratio exceeds 1. Dry soil changes character already near 1 MHz. These crossovers matter in practice: they are why submarines communicate at very low frequencies and why ground-penetrating radar works in dry soil but not in wet clay. Lesson 8 returns to them.

The second test is about size. A 60 Hz power line 1000 km long is about one fifth of a wavelength, so a continental grid is not a lumped circuit: utilities must treat long lines as transmission lines with travelling waves, which lesson 9 develops. A 5 cm circuit board at 2.4 GHz is nearly half a wavelength across, so its traces radiate and must be designed as waveguiding structures. Your phone's antenna exists precisely because of this. On the other hand the kitchen wiring that feeds a lamp is a millionth of a wavelength long, and Kirchhoff's laws apply to it essentially exactly.

The quasistatic approximation

Between pure statics and full electrodynamics lies a useful middle ground. In the quasistatic approximation we keep Faraday's law, which is essential for transformers, motors and generators, but drop the displacement current in conductors and ignore retardation, treating fields at each instant as though they were static fields produced by the sources at that instant. Lessons 2 and 3 work largely in this regime. Circuit theory is its engineering form: each inductor obeys \(V = L\,dI/dt\) because of Faraday's law, each capacitor stores charge \(Q = CV\) as in electrostatics, and the wires between them are short compared with a wavelength. The approximation fails when frequencies are high or distances are large, and then the full theory, with waves, is required.

The experiment that confirmed it

Maxwell died in 1879 without seeing his waves produced. Heinrich Hertz did so in Karlsruhe between 1886 and 1888. He charged a pair of metal rods until a spark jumped the small gap between them; the spark let charge surge back and forth at a frequency of tens of megahertz, set by the capacitance and inductance of the rods. A few metres away, a loop of wire with its own small gap showed tiny sparks whenever the transmitter fired. Hertz measured the wavelength from standing waves formed by reflection from a metal sheet, found the speed equal to that of light, and showed that the waves reflected, refracted and could be polarized exactly as light does. Within fifteen years the same physics was carrying wireless telegraphy across the Atlantic, as lesson 8 recounts.

How this course is organized

Lessons 2 and 3 complete the equations: electromotive force, Ohm's law and Faraday's law, then inductance, magnetic energy and the displacement current. Lesson 4 writes the full set in vacuum and in matter, with the boundary conditions that link fields across surfaces. Lesson 5 develops the conservation laws, showing that the fields carry energy, described by the Poynting vector, and momentum, described by the Maxwell stress tensor. Lessons 6 to 9 treat waves: plane waves in vacuum, reflection and refraction at surfaces including the Fresnel equations, absorption and dispersion in conductors and plasmas, and guided waves in waveguides and transmission lines. Lessons 10 and 11 return to the sources, introducing the potentials, gauge freedom and the retarded potentials, then deriving the radiation from oscillating dipoles and accelerating charges, with antennas and synchrotron light as applications. The capstone shows that Maxwell's equations are already consistent with special relativity, and that electric and magnetic fields are two faces of one object.

The course assumes PHYS-381 and the waves and optics course PHYS-230. Some topics overlap with PHYS-230, which met electromagnetic waves, polarization and interference at an introductory level; here they are derived from Maxwell's equations. PHYS-271 covered special relativity, which the capstone uses. Every lesson includes runnable Python using only the standard library, so that every numerical claim can be checked.