Electromagnetism

Electromagnetism is the study of electric charges, currents, and the fields they produce — and the deep unity that binds electricity and magnetism into a single force of nature. It is the first field theory in physics, culminating in Maxwell’s equations, which describe all classical electromagnetic phenomena in four elegant partial differential equations. From the attraction between a balloon and a sweater to the propagation of light across the cosmos, electromagnetism governs an astonishing range of the physical world and provides the template upon which all subsequent field theories — including general relativity and the Standard Model — are modeled.

Electrostatics: Charges and Fields

Electrostatics deals with electric charges at rest and the fields they create. The fundamental law is Coulomb’s law (1785): two point charges $q_1$ and $q_2$ separated by distance $r$ exert a force on each other of magnitude

$F = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2}$

where $\epsilon_0 = 8.854 \times 10^{-12}$ F/m is the permittivity of free space. The force is attractive for unlike charges and repulsive for like charges. The electric field $\mathbf{E}$ at a point is defined as the force per unit test charge: $\mathbf{E} = \mathbf{F}/q$. For a point charge $q$ at the origin, $\mathbf{E} = (q / 4\pi\epsilon_0 r^2)\hat{\mathbf{r}}$.

The electric field obeys Gauss’s law, the first of Maxwell’s equations:

$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \qquad \text{(differential form)}, \qquad \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \qquad \text{(integral form)}$

where $\rho$ is the charge density and $Q_{\text{enc}}$ is the total enclosed charge. Gauss’s law is most powerful for symmetric charge distributions: it immediately gives the field of an infinite plane ($\mathbf{E} = \sigma / 2\epsilon_0$), an infinite line charge ($\mathbf{E} = \lambda / 2\pi\epsilon_0 r$), and a uniformly charged sphere (field identical to a point charge outside the sphere).

Because the electrostatic field is conservative ($\nabla \times \mathbf{E} = 0$), we can define the electric potential $V$ such that $\mathbf{E} = -\nabla V$. The potential satisfies Poisson’s equation, $\nabla^2 V = -\rho/\epsilon_0$, which reduces to Laplace’s equation $\nabla^2 V = 0$ in charge-free regions. These equations, together with boundary conditions, determine the field everywhere — a program that can be carried out analytically for simple geometries and numerically for complex ones.

Conductors in electrostatic equilibrium have $\mathbf{E} = 0$ inside and charge residing entirely on the surface. The surface acts as an equipotential, and the field just outside is perpendicular to the surface with magnitude $\sigma/\epsilon_0$. This leads to electrostatic shielding (the Faraday cage) and to the concept of capacitance: $C = Q/V$, measuring a conductor’s ability to store charge at a given potential. For a parallel-plate capacitor with area $A$ and separation $d$, $C = \epsilon_0 A / d$. Dielectric materials between the plates increase the capacitance by a factor of the relative permittivity $\kappa$, because their molecular dipoles partially cancel the applied field.

Magnetostatics and the Lorentz Force

Magnetism arises from moving charges. A charge $q$ moving with velocity $\mathbf{v}$ in a magnetic field $\mathbf{B}$ experiences the Lorentz force:

$\mathbf{F} = q\mathbf{v} \times \mathbf{B}$

Because the magnetic force is always perpendicular to $\mathbf{v}$, it does no work — it changes the direction of motion but not the speed. In a uniform field, a charged particle traces a circle (or helix) with cyclotron radius $r = mv / qB$. This is the operating principle of cyclotrons, mass spectrometers, and the confinement of plasma in tokamaks.

Steady currents produce magnetic fields governed by the Biot-Savart law and Ampere’s law. The Biot-Savart law gives the field $d\mathbf{B}$ from a current element $Id\boldsymbol{\ell}$ at distance $\mathbf{r}$:

$d\mathbf{B} = \frac{\mu_0}{4\pi}\frac{Id\boldsymbol{\ell} \times \hat{\mathbf{r}}}{r^2}$

where $\mu_0 = 4\pi \times 10^{-7}$ T$\cdot$m/A is the permeability of free space. For a long straight wire carrying current $I$, integration gives $B = \mu_0 I / 2\pi r$ — a result also obtainable from Ampere’s law, $\oint \mathbf{B} \cdot d\boldsymbol{\ell} = \mu_0 I_{\text{enc}}$, by exploiting the cylindrical symmetry.

A crucial fact about magnetic fields is that they have no sources: $\nabla \cdot \mathbf{B} = 0$ everywhere. There are no magnetic monopoles (isolated north or south poles). Every magnetic field line is a closed loop, and the total magnetic flux through any closed surface is zero. This is the second of Maxwell’s equations and reflects a deep asymmetry between electricity and magnetism that has motivated decades of experimental searches for monopoles — so far without success.

Maxwell’s Equations and Electromagnetic Waves

The crowning achievement of classical electromagnetism is James Clerk Maxwell’s unification of electricity and magnetism into a single, self-consistent theory. Maxwell’s four equations, in differential form, are:

$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$

$\nabla \cdot \mathbf{B} = 0$

$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$

The third equation is Faraday’s law of induction (1831): a changing magnetic flux induces an electric field. It is the principle behind electrical generators, transformers, and inductors. The direction of the induced EMF opposes the change in flux (Lenz’s law), consistent with energy conservation.

The fourth equation contains Maxwell’s critical insight: the displacement current term $\mu_0 \epsilon_0 \partial \mathbf{E}/\partial t$. Maxwell recognized in 1861 that Ampere’s original law ($\nabla \times \mathbf{B} = \mu_0 \mathbf{J}$) was inconsistent with charge conservation when currents are time-dependent. Adding the displacement current restored consistency and had a momentous consequence: in free space ($\rho = 0$, $\mathbf{J} = 0$), the equations predict self-sustaining electromagnetic waves. Taking the curl of Faraday’s law and substituting the Ampere-Maxwell law yields the wave equation:

$\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}$

The wave speed is $c = 1/\sqrt{\mu_0 \epsilon_0} = 3 \times 10^8$ m/s — precisely the measured speed of light. Maxwell concluded that light is an electromagnetic wave, unifying optics with electromagnetism in one of the great syntheses of physics. Heinrich Hertz confirmed the existence of electromagnetic waves experimentally in 1887, generating and detecting radio waves in his laboratory.

Energy, Momentum, and Radiation

Electromagnetic fields carry energy and momentum. The energy density stored in the fields is:

$u = \frac{1}{2}\left(\epsilon_0 E^2 + \frac{B^2}{\mu_0}\right)$

The energy flux is described by the Poynting vector, $\mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B}$, which gives the power per unit area transported by the fields. Together they satisfy the Poynting theorem — the conservation of electromagnetic energy:

$\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = -\mathbf{J} \cdot \mathbf{E}$

The right side represents the rate at which the fields do work on charges (Joule heating). For a plane electromagnetic wave, the time-averaged intensity is $\langle S \rangle = \frac{1}{2}\epsilon_0 c E_0^2$, and the wave carries momentum density $\mathbf{g} = \mathbf{S}/c^2$, leading to radiation pressure $P = \langle S \rangle / c$ — a tiny but measurable force that pushes comet tails away from the Sun and may one day propel solar sails.

Accelerating charges radiate electromagnetic energy. The power radiated by a nonrelativistic charge with acceleration $a$ is given by the Larmor formula:

$P = \frac{q^2 a^2}{6\pi\epsilon_0 c^3}$

An oscillating electric dipole — a charge oscillating back and forth — radiates power proportional to $\omega^4$, with an angular distribution $\propto \sin^2\theta$ (maximum in the plane perpendicular to the dipole, zero along its axis). This is the simplest model of a radio antenna and explains why the sky is blue: sunlight scattering off atmospheric molecules (Rayleigh scattering) goes as $\omega^4$, favoring short wavelengths.

Relativistic Electrodynamics

Maxwell’s equations are already consistent with special relativity — indeed, it was the apparent conflict between Maxwell’s equations and Galilean relativity that led Albert Einstein to formulate special relativity in 1905. The electric and magnetic fields are not separate entities but components of a single electromagnetic field tensor $F^{\mu\nu}$, a rank-2 antisymmetric tensor in four-dimensional spacetime. Under a Lorentz transformation, electric and magnetic fields mix into each other: a purely electric field in one frame has both electric and magnetic components in another.

Maxwell’s equations take a remarkably compact form in relativistic notation:

$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu, \qquad \partial_{[\lambda} F_{\mu\nu]} = 0$

where $J^\nu = (\rho c, \mathbf{J})$ is the four-current. The electromagnetic potential becomes a four-vector $A^\mu = (V/c, \mathbf{A})$, and the field tensor is $F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. Gauge invariance — the freedom to add the gradient of a scalar to $A^\mu$ without changing the physics — is a fundamental symmetry that, when generalized, becomes the organizing principle of the Standard Model of particle physics.

The relativistic formulation reveals that electromagnetism is the prototypical gauge theory. The requirement that the Lagrangian be invariant under local phase transformations of the charged-particle wave function necessarily implies the existence of a vector field $A^\mu$ coupling to the current — predicting the photon from symmetry alone. This insight, due to Hermann Weyl and later elevated by Yang and Mills, is the conceptual foundation of modern particle physics: the strong and weak nuclear forces are also gauge theories, differing from electromagnetism only in the symmetry group.