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)}$$ Electric and magnetic fields are independent, set up by their own sources and oblivious to each other. This is why introductory courses can treat electrostatics and magnetostatics as separate topics. In the **dynamic case**, the curl equations couple $\mathbf E$ and $\mathbf B$: a changing one creates a curling other. This is the source of all wave-propagation phenomena, including light. ## Why these four? The number four isn't arbitrary. Each of $\mathbf E$ and $\mathbf B$ is a 3-vector field (six scalar components total), and to determine a vector field you need *two* PDEs: one for divergence, one for curl, by the Helmholtz decomposition. Two fields × two equations = four, exactly the right number to specify $\mathbf E$ and $\mathbf B$ in space. The equations also satisfy a consistency check: $\nabla \cdot (\nabla \times \mathbf E) = 0$ identically, so taking $\nabla\cdot$ of Faraday gives $\partial_t(\nabla \cdot \mathbf B) = 0$. Once $\nabla \cdot \mathbf B = 0$ holds initially, it holds forever. Similarly the $\partial_t \mathbf D$ term in Ampère is what makes $\partial_t(\nabla\cdot\mathbf D) + \nabla\cdot\mathbf J = 0$, the charge continuity equation, automatic. ## Wave equation derivation Take Faraday's law and apply curl to both sides: $$\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 (transmission lines, antennas, most engineering EM) these phasor Maxwell equations are what you'll actually compute with.