Electromagnetism

Matter is held together by enormous electrical forces, but they are so perfectly balanced between positive protons and negative electrons that we never feel them. Magnetism is a subtler effect — in fact a relativistic consequence of electricity. When charges move, their electric fields as seen by a moving observer are altered into what we call a magnetic field. Electricity and magnetism are not two things but two aspects of a single, unified electromagnetic field.

To talk about fields that vary from point to point, we need a new calculus and a special operator. The gradient of a field points the way of steepest increase. The divergence measures how much a field spreads out from a point, like a source. The curl measures how much it swirls or circulates around a point. These three ideas are the mathematical vocabulary needed to write the laws of electromagnetism.

Differential laws describe what happens at each point; integral laws describe the overall behavior. Gauss's theorem says the total flux of a field flowing out of a closed surface equals the sum of all the little sources (the divergence) inside it. Stokes' theorem is the partner statement: the total circulation around a closed loop equals the sum of all the little swirls (the curl) on the surface it encloses.

The world of stationary charges runs on two simple laws. First, electric field lines start on positive charges and end on negative ones, so the total flux out of a surface tells you the charge inside (Gauss's law). Second, the electrostatic field has no swirl (its curl is zero), which means we can define an electric potential, or voltage. The whole subject of electrostatics lives inside those two statements.

Gauss's law is a powerful shortcut for finding electric fields where there is symmetry, avoiding hard integrals. For a uniformly charged sphere, symmetry says the field must point straight out; draw an imaginary spherical surface and Gauss's law instantly shows the outside field is the same as if all the charge sat at the center. It also proves the field inside a hollow charged conductor is exactly zero — a very sensitive test of the inverse-square law.

When conductors are present, charges shuffle around until the surface is all at one potential, which makes problems tricky. A clever fix is the 'method of images': to find the field of a charge near a conducting plane, pretend the plane is gone and place an imaginary mirror-image charge on the other side. The field in the real region comes out exactly right. At sharp points the field grows very strong — which is why lightning rods are pointed.

The equations of electrostatics turn up far beyond charges. For two-dimensional problems there is a powerful method using functions of a complex variable, where the real and imaginary parts of any smooth function automatically solve the field equations. We also see the same equations in motion — as in plasma oscillations, where displaced electrons in an ionized gas are pulled back and oscillate at a characteristic 'plasma frequency.'

It takes work to push charges together against their repulsion, and that work is stored as potential energy. But where is the energy? A very useful idea is that it is not in the charges but in the electric field they create, with an energy density proportional to the square of the field strength. This picture — energy living in the field — is essential for understanding how electromagnetic waves carry energy through empty space.

On a clear day there is a downward electric field of about 100 volts per meter in the air — the Earth itself is negatively charged. This drives a small current that should drain the charge in about half an hour, so what keeps the Earth charged? Thunderstorms. They act like giant batteries, carrying negative charge down to the ground in lightning and positive charge up to the high atmosphere, maintaining a global electrical circuit.

Put an insulating material — a dielectric — into an electric field and the field inside it weakens. This is because the field polarizes the atoms, stretching them into tiny dipoles whose own field opposes the original. How much the field is reduced is a property called the dielectric constant. This is exactly why a capacitor can store more charge when filled with a dielectric material.

Magnetostatics studies the magnetic fields of steady currents. Two laws rule it: magnetic field lines never start or stop but form closed loops, and the circulation of the magnetic field around a loop is set by the total current passing through it (Ampere's law). And underneath it all, magnetism is a relativistic effect of electricity — a magnetic field is what an electric field looks like to a moving observer.

Just as the electric field comes from a voltage, the magnetic field can be derived from a 'vector potential.' Is it a real field or just a math convenience? In classical physics you could argue either way, but quantum mechanics settles it: it is real. The Aharonov-Bohm effect shows an electron can be affected by the vector potential even where the magnetic field is zero, shifting the phase of its wave. The vector potential is more fundamental than the magnetic field itself.

Faraday discovered that a changing magnetic field creates an electric field — the principle of induction. Move a magnet near a wire loop, or change a nearby current, and a current is induced. This is the basis of generators, which turn motion into electricity, and transformers, which change voltages. The induced current always flows so as to oppose the very change that made it (Lenz's law).

This is the great moment of synthesis. Maxwell noticed the known laws of electricity and magnetism were inconsistent with conservation of charge, and fixed them by adding a new term, the 'displacement current.' That one addition had a stunning payoff: the equations now predicted self-propagating electromagnetic waves traveling at a speed he could calculate — exactly the known speed of light. In a stroke, light was revealed to be an electromagnetic wave.

Like mechanics, all of electrodynamics can be summed up in one powerful principle: least action. The motion of a charged particle and the behavior of the field can both be found by choosing the configuration that makes a quantity called the 'action' stationary. This abstract, beautiful idea is the foundation of modern field theory, including quantum electrodynamics — and it suggests Nature is, in some deep sense, economical.

The laws of electromagnetism let us analyze alternating-current circuits. For smoothly oscillating voltages and currents there is a beautiful trick using complex numbers: each component — resistor, capacitor, inductor — gets a complex 'impedance.' This turns the circuit's differential equations into simple algebra, making even complicated networks tractable. And resonance, so important in mechanics, shows up here all over again.

To send high-frequency waves like microwaves from place to place you cannot just use wires — they would act as antennas and radiate the energy away. Instead you use a hollow metal pipe, a waveguide, where the waves bounce along the conducting walls. A waveguide only carries waves above a certain cutoff frequency set by its size; below that, waves quickly die out. Waveguides are the 'plumbing' of microwave engineering.