Maxwell's Equations and Boundary Conditions

Syllabus: Maxwell's Equations: Differential and integral forms and their interpretation, boundary conditions, wave equation, Poynting vector.

7 hand-picked visual resources on this topic, 3 of them little-known gems — under each card, look for See it. Every link is checked against the live site before it ships.

The four equations, both forms

The displacement-current term $\partial D/\partial t$ is Maxwell's own addition and the reason waves exist at all. In statics it vanishes and the electric and magnetic sets decouple — which is exactly what a question means when it says 'static fields'.

$\nabla\cdot D=\rho_v\ \Leftrightarrow\ \oint D\cdot dS=Q_{enc}$
Gauss's law
$\nabla\cdot B=0\ \Leftrightarrow\ \oint B\cdot dS=0$
no magnetic monopoles
$\nabla\times E=-\dfrac{\partial B}{\partial t}\ \Leftrightarrow\ \oint E\cdot dl=-\dfrac{d\Phi}{dt}$
$\nabla\times H=J+\dfrac{\partial D}{\partial t}\ \Leftrightarrow\ \oint H\cdot dl=I_{enc}+I_d$
$\nabla\cdot J=-\dfrac{\partial\rho_v}{\partial t}$
continuity — implied by the set, not independent

Where marks are lost

A field given as a candidate $B$ must satisfy $\nabla\cdot B=0$ ; a candidate static $E$ must satisfy $\nabla\times E=0$

See it

Boundary conditions

Tangential $E$ and normal $B$ are always continuous. The other two jump by whatever free surface charge or current sits on the interface — at a perfect conductor, that jump is the whole answer.

$E_{t1}=E_{t2},\qquad B_{n1}=B_{n2}$

Wave equation and the Poynting vector

$S=E\times H$ points where the power flows and its magnitude is the power density in W/m². For a uniform plane wave it lies along the direction of propagation, and its time average is what a power meter reads.

$\nabla^2E=\mu\varepsilon\dfrac{\partial^2E}{\partial t^2}$

Plane Waves, Media and Interfaces →

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