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Maxwell’s equations are the foundational postulates of classical electrodynamics, optics, and electric circuits. At a high level, they do not just describe electric and magnetic fields in isolation, but reveal them as unified manifestations of a single underlying electromagnetic field, intimately connected to the geometry of spacetime.

The Principia

In standard SI units, the differential form of Maxwell’s equations defines the relationship between the electric field $\mathbf{E}$, the magnetic field $\mathbf{B}$, the charge density $\rho$, and the current density $\mathbf{J}$.

  1. Gauss’s Law: Describes how electric charges produce electric fields. The electric flux leaving a volume is proportional to the charge inside.

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$$

  1. Gauss’s Law for Magnetism: States there are no magnetic monopoles; magnetic field lines are continuous loops.

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

  1. Faraday’s Law of Induction: A time-varying magnetic field induces a circulating electric field.

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

  1. Ampère-Maxwell Law: Magnetic fields are generated by both electrical currents and time-varying electric fields (the displacement current).

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$$

(Note: $\varepsilon_0$ is the vacuum permittivity, $\mu_0$ is the vacuum permeability, and the speed of light is $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$).

Potentials and Gauge Invariance

To simplify the coupling of these fields to quantum mechanics and relativity, we express $\mathbf{E}$ and $\mathbf{B}$ in terms of the scalar potential $\phi$ and the vector potential $\mathbf{A}$.

Because $\nabla \cdot \mathbf{B} = 0$, we can define $\mathbf{B}$ as the curl of a vector field:

$$\mathbf{B} = \nabla \times \mathbf{A}$$

Substituting this into Faraday’s law yields $\nabla \times (\mathbf{E} + \frac{\partial \mathbf{A}}{\partial t}) = 0$, meaning the quantity in parentheses can be written as the gradient of a scalar field:

$$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}$$

Gauge Freedom: The physical fields $\mathbf{E}$ and $\mathbf{B}$ are invariant under the local gauge transformations:

  • $\mathbf{A} \rightarrow \mathbf{A} + \nabla \lambda$
  • $\phi \rightarrow \phi - \frac{\partial \lambda}{\partial t}$

Where $\lambda(\mathbf{x}, t)$ is an arbitrary scalar function. This $U(1)$ gauge symmetry is the fundamental principle from which quantum electrodynamics (QED) is derived. Imposing the Lorenz gauge ($\nabla \cdot \mathbf{A} + \frac{1}{c^2}\frac{\partial \phi}{\partial t} = 0$) decouples the wave equations for the potentials.

The Covariant Formulation (Relativistic)

To make the Lorentz invariance of Maxwell’s equations explicit, we upgrade our 3D quantities to 4D tensors in Minkowski spacetime with metric $\eta_{\mu\nu} = \text{diag}(-1, 1, 1, 1)$.

1. Four-Vectors

  • Four-Current: Combines charge density and 3-current.

$$J^\mu = (c\rho, \mathbf{J})$$

  • Four-Potential: Combines the scalar and vector potentials.

$$A^\mu = \left(\frac{\phi}{c}, \mathbf{A}\right)$$

2. The Electromagnetic Field Tensor

The physical fields are not four-vectors, but rather the components of an antisymmetric rank-2 tensor, the Faraday tensor $F^{\mu\nu}$, defined as the four-curl of the four-potential:

$$ F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu $$

In matrix representation (SI units), this elegantly encodes both fields:

$$ F^{\mu\nu} = \begin{pmatrix} 0 & E_x/c & E_y/c & E_z/c \\\ -E_x/c & 0 & B_z & -B_y \\\ -E_y/c & -B_z & 0 & B_x \\\ -E_z/c & B_y & -B_x & 0 \end{pmatrix} $$

3. Covariant Maxwell’s Equations

The four classical equations reduce to just two tensor equations:

The Inhomogeneous Equation (Gauss & Ampère-Maxwell):

$$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$$

This states that the four-divergence of the field tensor is proportional to the four-current.

The Homogeneous Equation (Faraday & Gauss for Magnetism): $$\partial_\lambda F_{\mu\nu} + \partial_\mu F_{\nu\lambda} + \partial_\nu F_{\lambda\mu} = 0$$

This is the Bianchi identity. It can also be written using the dual tensor $\tilde{F}^{\mu\nu} = \frac{1}{2}\epsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}$ (where $\epsilon$ is the Levi-Civita symbol) as:

$$\partial_\mu \tilde{F}^{\mu\nu} = 0$$

Fundamental Consequences

  • Continuity Equation: Taking the four-divergence of the inhomogeneous equation yields $\partial_\nu \partial_\mu F^{\mu\nu} = \mu_0 \partial_\nu J^\nu$. Since $F^{\mu\nu}$ is antisymmetric and partial derivatives commute, the LHS is zero, mandating the local conservation of charge:

$$\partial_\mu J^\mu = 0 \implies \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0$$

  • Electromagnetic Waves: In a vacuum ($J^\mu = 0$), under the Lorenz gauge ($\partial_\mu A^\mu = 0$), the inhomogeneous equation simplifies to the 4D wave equation:

$$\square A^\mu = 0$$

(Where $\square = \partial_\mu \partial^\mu$ is the d’Alembertian operator). This proves that electromagnetic disturbances propagate at $c$, independent of the inertial frame.

  • Lorentz Force Law: The relativistic dynamics of a particle with charge $q$ and 4-velocity $U^\mu$ interacting with the field is cleanly given by:

$$\frac{dP^\mu}{d\tau} = q F^{\mu\nu} U_\nu$$