Maxwell’s equations are the four PDEs that completely describe classical electromagnetism — the relations among the Electric field , magnetic flux density , electric flux density , magnetic field intensity , Charge density , and current density .
In differential form:
\nabla \cdot \mathbf D &= \rho_v \quad \text{(Gauss's law)} \\ \nabla \cdot \mathbf B &= 0 \quad \text{(no magnetic monopoles)} \\ \nabla \times \mathbf E &= -\frac{\partial \mathbf B}{\partial t} \quad \text{(Faraday's law)} \\ \nabla \times \mathbf H &= \mathbf J + \frac{\partial \mathbf D}{\partial t} \quad \text{(Ampère with displacement current)} \end{align}$$ In integral form: $$\begin{align} \oint_S \mathbf D \cdot d\mathbf s &= Q_{\text{enc}} \\ \oint_S \mathbf B \cdot d\mathbf s &= 0 \\ \oint_C \mathbf E \cdot d\mathbf l &= -\int_S \frac{\partial \mathbf B}{\partial t} \cdot d\mathbf s \\ \oint_C \mathbf H \cdot d\mathbf l &= I_{\text{enc}} + \int_S \frac{\partial \mathbf D}{\partial t} \cdot d\mathbf s \end{align}$$ These are completed by the **constitutive relations** linking $\mathbf D, \mathbf E$ and $\mathbf B, \mathbf H$ through material parameters: $$\mathbf D = \epsilon \mathbf E, \qquad \mathbf B = \mu \mathbf H, \qquad \mathbf J = \sigma \mathbf E.$$ For linear, isotropic, homogeneous media. Electromagnetics assumes these unless explicitly noted. ## What each one says **[[Gauss's law]]** ($\nabla \cdot \mathbf D = \rho_v$): electric flux lines start on positive free charges and end on negative ones. Equivalently, the divergence of $\mathbf D$ at a point equals the local free charge density. **Gauss's law for magnetism** ($\nabla \cdot \mathbf B = 0$): no magnetic monopoles. Every magnetic field line is a closed loop. **[[Faraday's law]]** ($\nabla \times \mathbf E = -\partial_t \mathbf B$): a time-varying magnetic field generates a curling (non-conservative) electric field. The closed-loop integral of $\mathbf E$ is no longer zero in dynamic problems. **[[Ampère's law]] with [[Displacement current]]** ($\nabla \times \mathbf H = \mathbf J + \partial_t \mathbf D$): currents *and* time-varying electric fields source curling $\mathbf H$. The $\partial_t \mathbf D$ term is Maxwell's correction — without it the equations would not predict EM waves and would violate charge conservation. ## Static vs dynamic In the **static case**, $\partial_t = 0$ everywhere, and the equations decouple: $$\begin{align} \nabla \cdot \mathbf D &= \rho_v \\ \nabla \times \mathbf E &= 0 \end{align} \qquad \text{(electrostatics)}$$ $$\begin{align} \nabla \cdot \mathbf B &= 0 \\ \nabla \times \mathbf H &= \mathbf J \end{align} \qquad \text{(magnetostatics)}$$ Electric and magnetic fields are independent — set up by their own sources, oblivious to each other. This is why introductory courses can treat electrostatics and magnetostatics as separate topics. In the **dynamic case**, the curl equations couple $\mathbf E$ and $\mathbf B$: a changing one creates a curling other. This is the source of all wave-propagation phenomena, including light. ## Why these four? The number four isn't arbitrary. Each of $\mathbf E$ and $\mathbf B$ is a 3-vector field — six scalar components total — and to determine a vector field you need *two* PDEs (one for divergence, one for curl, by the Helmholtz decomposition). Two fields × two equations = four. The structure isn't accidental; it's exactly the right number to specify $\mathbf E$ and $\mathbf B$ in space. The equations also satisfy a consistency check: $\nabla \cdot (\nabla \times \mathbf E) = 0$ identically, so taking $\nabla\cdot$ of Faraday gives $\partial_t(\nabla \cdot \mathbf B) = 0$ — once $\nabla \cdot \mathbf B = 0$ holds initially, it holds forever. Similarly the $\partial_t \mathbf D$ term in Ampère is what makes $\partial_t(\nabla\cdot\mathbf D) + \nabla\cdot\mathbf J = 0$ — the charge continuity equation — automatic. ## Wave equation derivation Take Faraday's law and apply curl to both sides: $$\nabla \times (\nabla \times \mathbf E) = -\frac{\partial}{\partial t}(\nabla \times \mathbf B).$$ Use the [[Curl of curl identity]] on the left: $\nabla(\nabla\cdot\mathbf E) - \nabla^2 \mathbf E$. In a charge-free region, $\nabla \cdot \mathbf E = 0$, leaving $-\nabla^2 \mathbf E$. For the right side in source-free vacuum, use $\nabla \times \mathbf B = \mu_0 \epsilon_0 \partial_t \mathbf E$ (Ampère's with $\mathbf J = 0$). Substitute: $$-(-\nabla^2 \mathbf E) = -\mu_0 \epsilon_0 \frac{\partial^2 \mathbf E}{\partial t^2},$$ i.e., $$\nabla^2 \mathbf E - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf E}{\partial t^2} = 0.$$ This is the **wave equation** for $\mathbf E$, with propagation speed $$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}.$$ Same derivation starting from Ampère's law gives the same wave equation for $\mathbf B$. The two fields propagate together as a self-sustaining EM wave. In a region *with* sources ($\rho_v \neq 0$, $\mathbf J \neq 0$), the same procedure gives a **source-driven wave equation**: $$\nabla^2 \mathbf E - \mu_0\epsilon_0\frac{\partial^2 \mathbf E}{\partial t^2} = \mu_0\frac{\partial \mathbf J}{\partial t} + \frac{1}{\epsilon_0}\nabla\rho_v.$$ The right side is what makes accelerating charges radiate — antennas, dipoles, every EM emitter. The source-free version above is the special case in regions where you've already left the antenna behind and the wave is propagating on its own. ## In phasor form For time-harmonic fields $\mathbf E(\mathbf r, t) = \text{Re}[\tilde{\mathbf E}(\mathbf r) e^{j\omega t}]$, replace $\partial_t$ with $j\omega$: $$\begin{align} \nabla \cdot \tilde{\mathbf D} &= \tilde\rho_v \\ \nabla \cdot \tilde{\mathbf B} &= 0 \\ \nabla \times \tilde{\mathbf E} &= -j\omega \tilde{\mathbf B} \\ \nabla \times \tilde{\mathbf H} &= \tilde{\mathbf J} + j\omega \tilde{\mathbf D} \end{align}$$ In sinusoidal steady state — the realm of transmission lines, antennas, and most engineering EM — these phasor Maxwell equations are what you'll actually compute with. See [[Phasor]] and [[Phasor transform]].