Maxwell's Equations

This section summarizes the complete set of four equations that govern classical electrodynamics, uniting electricity, magnetism, and electromagnetic waves.

1. The Complete Set

Electrodynamics is governed by four partial differential equations (Maxwell's) and one force law (Lorentz). Together, they describe everything from light waves to electric motors.

The Force Law

This tells us how fields affect charges:

\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})

2. The Ampère-Maxwell Correction

The field-to-circuit bridge used charge conservation to explain currents at nodes and on capacitor plates. We now return to that same charging process at the field level.

In steady circuits, current flows in a complete loop. But consider a charging capacitor: current flows into one plate and out of the other, but nothing (no charges) physically crosses the gap.

This creates a paradox: if there is no current in the gap, Ampère's Law says there should be no magnetic field there. But experiments show the field does exist. Maxwell realized that as the plates charge, the Electric Field between them is changing, and this change acts exactly like a current.

The Full Ampère-Maxwell Law

Maxwell's correction to Ampère's Law, incorporating the displacement current density ($J_d$):

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

Where the term $\epsilon_0 \frac{\partial \vec{E}}{\partial t}$ is the Displacement Current.

Deep Dive: How Maxwell Fixed Ampère's Law

3. The General Equations (Vacuum)

In free space, these are the fundamental relationships.

\begin{align} \text{(i)} \quad & \nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0} \\ \text{(ii)} \quad & \nabla \cdot \vec{B} = 0 \\ \text{(iii)} \quad & \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \\ \text{(iv)} \quad & \nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t} \end{align}

(i) Gauss's Law: Charges create divergence in E.

(ii) No Monopoles: Magnetic field lines are closed loops.

(iii) Faraday's Law: Changing B creates curling E.

(iv) Ampère-Maxwell: Currents (and changing E) create curling B.

Deep Dive: Magnetic Monopoles & Symmetry

4. Maxwell's Equations in Matter

In materials, we abstract away the trillions of bound charges and currents into auxiliary fields $\vec{D}$ and $\vec{H}$.

Auxiliary Definitions

\vec{D} \equiv \epsilon_0 \vec{E} + \vec{P}, \quad \vec{H} \equiv \frac{1}{\mu_0}\vec{B} - \vec{M}

Macroscopic Equations

\begin{align} \nabla \cdot \vec{D} &= \rho_f \\ \nabla \cdot \vec{B} &= 0 \\ \nabla \times \vec{E} &= -\frac{\partial \vec{B}}{\partial t} \\ \nabla \times \vec{H} &= \vec{J}_f + \frac{\partial \vec{D}}{\partial t} \end{align}

Constitutive Relations (Linear Media)

To solve these, we need to know how the material responds to fields:

\vec{P} = \epsilon_0 \chi_e \vec{E}, \quad \vec{M} = \chi_m \vec{H}

Or simply: $\vec{D} = \epsilon \vec{E}$ and $\vec{H} = \frac{1}{\mu} \vec{B}$.

Deep Dive: Maxwell's Equations in Matter

5. From Maxwell Fields to Lumped Circuits

Maxwell's equations describe continuous electric and magnetic fields throughout space. Circuit theory does not introduce separate fundamental objects; it compresses an appropriate field problem into a few lumped quantities: voltage $V$, current $I$, resistance $R$, capacitance $C$, and inductance $L$.

When the Compression Works

The lumped-element approximation is appropriate when the circuit dimensions are much smaller than the relevant electromagnetic wavelength. Equivalently, the propagation time across the circuit is much shorter than the timescale on which the signal changes. Then the different parts of a component respond essentially together, and its distributed fields can be represented by a small number of terminal quantities. This is a scale comparison, not a universal numerical cutoff.

Voltage, Current, and Resistance

In an electrostatic or sufficiently quasistatic region, the relevant electric field is approximately conservative, so a single-valued scalar potential can describe the potential difference between two points:

V_B-V_A=-\int_A^B \mathbf{E}\cdot d\mathbf{l}.

This is the field meaning of voltage. It is not the same as an EMF, which is work per unit charge supplied by non-electrostatic forces or by an induced field around a loop. For a stationary loop linked by changing magnetic flux, Faraday's law gives $\oint\mathbf{E}\cdot d\mathbf{l}=-d\Phi_B/dt$; then a simple electrostatic potential picture alone is insufficient.

Circuit current is the current-density field integrated across a conductor cross-section:

I=\int_A \mathbf{J}\cdot d\mathbf{a}.

A material relation, not a Maxwell equation, supplies resistance. In an ohmic material, $\mathbf{J}=\sigma\mathbf{E}$. With an approximately quasistatic field distribution and a specified geometry, this local relation becomes the lumped relation $V=IR$. Thus $R$ summarizes conductivity, shape, and the field pattern in that component; it is not a fundamental object of nature.

Field-Energy Elements: Capacitors and Inductors

A capacitor begins with charge separation. That charge produces an electric field; Gauss's law and the field integral determine the potential difference; the geometry and material response are then summarized by $Q=CV$. For constant $C$,

I=\frac{dQ}{dt}=C\frac{dV}{dt}, \qquad u_E=\frac{1}{2}\epsilon E^2.

A capacitor is therefore an electric-field energy-storage system, not merely a symbol that obeys a circuit rule. As it charges, $dQ/dt$ accompanies a changing electric field in the gap; the Ampère-Maxwell correction is the field-level reason that this capacitor branch current is represented consistently on either side of the gap.

An inductor follows the complementary chain: current produces a magnetic field, the field produces linked magnetic flux, and changing current changes that flux. Faraday's law then produces an induced EMF. For a fixed geometry in the linear regime, the flux linkage $\lambda$ is proportional to current:

\lambda=LI, \qquad \mathcal{E}_{\mathrm{ind}}=-\frac{d\lambda}{dt}, \qquad v_L=L\frac{dI}{dt}.

Here $\lambda$ is the total flux linkage (for one turn, it is the flux). The last relation uses the passive sign convention: it is the terminal voltage drop, while the induced EMF has the opposite reference direction and opposes the change. An inductor is a magnetic-field energy-storage system.

Why KCL and KVL Work

Kirchhoff's current law is rooted in charge conservation, not in an independent circuit principle:

\nabla\cdot\mathbf{J}=-\frac{\partial\rho}{\partial t} \qquad\Longrightarrow\qquad \sum I_{\mathrm{in}}-\sum I_{\mathrm{out}}=\frac{dQ_{\mathrm{node}}}{dt}.

Ordinary lumped KCL, $\sum I=0$, applies when a small node has no significant unresolved net charge accumulation, or when every relevant branch current has been represented consistently. The familiar current balance is therefore a compact consequence of the continuity equation.

Kirchhoff's voltage law starts from Faraday's law:

\oint \mathbf{E}\cdot d\mathbf{l}=-\frac{d\Phi_B}{dt}.

If changing magnetic flux linked with the circuit loop is negligible, the quasistatic electric-field circulation is approximately zero, leading to the familiar sum of signed voltage rises and drops. If significant linked flux is present, a naive sum of electrostatic voltage drops is incomplete. The induced EMF can instead be represented explicitly as a source or element term, as in the induction lesson, so consistent circuit equations can still be written.

Where the Approximation Ends

As spatial or temporal field structure becomes important, a single current or voltage can no longer represent an entire component. Propagation delay, transmission-line behavior, parasitic capacitance and inductance, radiation from antennas, and skin-effect current crowding are common signs that a distributed electromagnetic description is needed.

Summary Map

Circuit quantity/law	Field origin	Main approximation	Failure or extension
$V$	Line integral of $\mathbf{E}$	Relevant $\mathbf{E}$ is approximately conservative	Use EMF and the full field description for induced circulation
$I$	$I=\int\mathbf{J}\cdot d\mathbf{a}$	One branch current represents the cross-section	Track nonuniform or propagating current density
$R$	Local conduction $\mathbf{J}=\sigma\mathbf{E}$	Linear material response and quasistatic field pattern	Nonlinear, frequency-dependent, or skin-effect response
$C$	Charge, $\mathbf{E}$, Gauss's law, and field energy	Fields are captured by one terminal-pair capacitance	Parasitic or distributed capacitance
$L$	$\mathbf{B}$, flux linkage, and Faraday's law	Flux linkage is represented by one $L$	Mutual, parasitic, radiating, or distributed inductance
KCL	Charge conservation and $\nabla\cdot\mathbf{J}=-\partial\rho/\partial t$	Negligible unresolved node-charge accumulation	Include charge accumulation and relevant capacitive/displacement effects
KVL	Electric-field circulation and Faraday's law	Negligible unrepresented changing linked magnetic flux	Include induced EMF or use a distributed electromagnetic model

Circuit theory is not a separate set of mysterious laws. It is a compact approximation to electromagnetic field theory that becomes extremely powerful when the circuit is sufficiently lumped. When that condition no longer holds, the next step is to return to the distributed fields that lead naturally to electromagnetic waves.