Chapter 11: Maxwell & Vector Calculus
1861–1873
The Mathematical Tools
Before Maxwell could write his equations, the mathematical language had to exist. Calculus gave derivatives and integrals for scalar functions of one variable. But electromagnetic fields are vector fields: at each point in space, they have both a magnitude and a direction. The calculus of vector fields required new operations.
The modern notation was not Maxwell's own. It was assembled by Josiah Willard Gibbs and Oliver Heaviside in the 1880s, who extracted the three key operations from Hamilton's quaternion calculus and gave them their current compact form:
Points in the direction of steepest increase of a scalar field. The electric field is the negative gradient of the electric potential.
Measures the net outward flux of a vector field per unit volume. Positive divergence means the point is a source; negative means a sink.
Measures the rotation or circulation of a vector field. A magnetic field circulates around a current; a nonzero curl means the field is not conservative.
Maxwell's Equations
James Clerk Maxwell (1831–1879) synthesized decades of experimental work by Faraday, Ampère, and others into four equations. In their differential form, using SI units:
Gauss's Law (Electric)
Electric field diverges from charge sources. \(\rho\) is the charge density. Electric monopoles exist.
Gauss's Law (Magnetic)
Magnetic field has no divergence. There are no magnetic monopoles — field lines always close on themselves.
Faraday's Law
A changing magnetic field creates a circulating electric field. The principle behind transformers, generators, and induction.
Ampère–Maxwell Law
A current \(\mathbf{J}\) or a changing electric field creates a circulating magnetic field. The second term is Maxwell's crucial addition.
The Displacement Current: A Mathematical Prediction
The second term in Ampère's law —\(\mu_0\varepsilon_0\,\partial\mathbf{E}/\partial t\) — was Maxwell's own addition, made not from experiment but from mathematical consistency. The original Ampère's law violated conservation of charge when applied to capacitors. Maxwell added a “displacement current” to fix this.
The consequences were immediate. Taking the curl of Faraday's law and substituting the Ampère–Maxwell law, in free space (\(\rho=0,\,\mathbf{J}=0\)):
\( \nabla^2\mathbf{E} = \mu_0\varepsilon_0\,\frac{\partial^2\mathbf{E}}{\partial t^2} \)
This is a wave equation. The wave speed is:
\( c = \frac{1}{\sqrt{\mu_0\varepsilon_0}} \approx 3\times10^8\;\text{m/s} \)
Maxwell computed this from the known values of \(\mu_0\) and\(\varepsilon_0\) and recognized it as the speed of light. He wrote in 1865: “We can scarcely avoid the inference that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” Electricity, magnetism, and light were unified in a single mathematical stroke.
The Prototype for All Field Theories
Maxwell's equations are far more than a description of electromagnetism. They established the template for how modern physics works:
Not particles acting at a distance, but fields filling space that carry energy and momentum. This is the ontology of all modern physics.
Physical laws are local: what happens at a point depends only on the field and its derivatives at that point, not on distant regions.
Field equations generically admit wave solutions. Electromagnetic waves, gravitational waves, and quantum mechanical wavefunctions all emerge this way.
Maxwell's equations are invariant under \(\mathbf{A} \to \mathbf{A} + \nabla\chi\), \(\phi \to \phi - \partial_t\chi\). This gauge freedom, initially seen as a mathematical redundancy, became the organizing principle of the Standard Model.
Einstein's Debt to Maxwell
Special relativity was discovered by asking what transformations leave Maxwell's equations invariant. The answer was the Lorentz group, not the Galilean group. Newton's mechanics needed modification; Maxwell's equations did not. The speed of light appeared as a constant in the equations — Einstein took this seriously and built a new mechanics around it.