Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Two of the four equations survive unchanged. There are still no magnetic charges when the fields vary, so ④ stays as it is, and the flux of $D$ out of a closed surface still counts the enclosed charge at every instant, so ③ stays as well. The two curl equations, on the other hand, each receive one new term:

(1) (2) \nabla \times E + \frac{\partial B}{\partial t} = 0, \nabla \times H - \frac{\partial D}{\partial t} = J .

The first is the Faraday–Maxwell equation: a magnetic field that changes in time has an electric field circulating around it. The second is the generalised Ampère law (Ampère–Maxwell equation): an electric flux density that changes in time has a magnetic field circulating around it, exactly as a current does. The two new terms have a very different history. The term $\partial B / \partial t$ was discovered experimentally by Faraday in 1831, thirty years before Maxwell wrote the equations. The term $\partial D / \partial t$ was never measured before it was written down: Maxwell added it in 1862 because the equations are inconsistent without it, as Section 5.2 shows. Table 5.1.1 collects the four equations in both forms.

Table 5.1.1. The four Maxwell equations at a glance, in differential and in integral form. The integral forms hold for any closed surface $S$ enclosing the volume $V$ , and for any closed curve $C$ bounding the (fixed) open surface $S$ , with $d l$ and $d S$ related by the right-hand rule.

Name	Differential form	Integral form
① Faraday–Maxwell	$\nabla \times E = - \frac{\partial B}{\partial t}$	$\oint_{C} E \cdot d l = - \frac{d}{d t} \int_{S} B \cdot d S$
② Generalised Ampère (Ampère–Maxwell)	$\nabla \times H = J + \frac{\partial D}{\partial t}$	$\oint_{C} H \cdot d l = \int_{S} (J + \frac{\partial D}{\partial t}) \cdot d S$
③ Gauss's law	$\nabla \cdot D = ρ_{v}$	$\oint_{S} D \cdot d S = \int_{V} ρ_{v} d V = Q_{enc}$
④ Conservation of magnetic flux	$\nabla \cdot B = 0$	$\oint_{S} B \cdot d S = 0$

Where each was built:

① $\nabla \times E = 0$ in Section 3.3, the $\partial B / \partial t$ term in Section 5.3,
② $\nabla \times H = J$ in Section 4.1, the $\partial D / \partial t$ term in Section 5.2,
③ Section 3.2, from Coulomb's law of Section 3.1, with $D = ε E$ in Section 3.4,
④ Section 4.1, unchanged by time variation.

Maxwell's equations

In a medium described by $D = ε E$ and $B = μ H$ , the electromagnetic field obeys, at every point and at every instant,

\nabla \times E \nabla \times H \nabla \cdot D \nabla \cdot B = - \frac{\partial B}{\partial t}, = J + \frac{\partial D}{\partial t}, = ρ_{v}, = 0. (5.1.1)

The symbols and their units are $E$ electric field $[V/m]$ , $H$ magnetic field $[A/m]$ , $D$ electric flux density $[C/ m^{2}]$ , $B$ magnetic flux density $[T]$ or $[Wb/ m^{2}]$ , $ρ_{v}$ charge density $[C/ m^{3}]$ , $J$ current density $[A/ m^{2}]$ .

In plain words

Charges make electric flux ( $\nabla \cdot D = ρ_{v}$ ). Nothing makes magnetic flux begin or end ( $\nabla \cdot B = 0$ ). Currents and changing electric fields make magnetic fields circulate ( $\nabla \times H$ ). Changing magnetic fields make electric fields circulate ( $\nabla \times E$ ). The two "changing" statements are the whole difference between statics and electrodynamics.

5.1.2 What changes when the fields vary in time