EM.3 · Maxwell waves, energy flow and boundary conditions

A charged capacitor can change its electric field across a gap even though no conduction current crosses the gap.

Prerequisites: 13, 29, 47.

• Use displacement current 位移电流 and material wave-speed relations
• Determine wave-field directions and radiated energy flow
• Apply Maxwell boundary conditions without confusing field components

Ampere–Maxwell law in vacuum is ∮B·dl=μ0(I_conduction+ε0 dΦE/dt). The added displacement-current term depends on the time rate of electric flux through the chosen surface. It ensures consistent results for a loop whose spanning surface either crosses a capacitor wire or passes between the plates. Displacement current is not magnetic flux or an integral over earlier electric flux. For a uniform plate field, the gap contribution is ε0A dE/dt; signs follow the oriented surface.

In a homogeneous, linear, lossless dielectric 电介质 with permeability μ and permittivity ε, Maxwell equations give wave speed v=1/sqrt(με) and refractive index n=c/v=sqrt(μrεr). For nonmagnetic material μr≈1, so v=c/sqrt(εr). Use the stated frequency-dependent material constants when dispersion matters; a static dielectric constant need not describe every optical frequency. Vacuum waves have E/B=c; in this simple medium the relation is E/B=v.

For a plane wave travelling along unit vector n, B=(n×E)/v. Both fields are perpendicular to propagation and to each other. Phase kz−ωt propagates toward +z, while kz+ωt propagates toward −z. If E is along x+y and propagation is +z, B is along −x+y, because z×x=y and z×y=−x. The Poynting vector 坡印廷矢量 S=E×H (E×B/μ in this medium) points along energy transport. In the radiation zone of an accelerating charge, energy flux is outward from the source, not necessarily in the instantaneous direction of charge motion.

Maxwell’s divergence equation ∇·B=0 gives continuity of the normal B component across an interface: a thin pillbox has no magnetic charge inside. The tangential H jump is related to surface current; it need not vanish. Under an ideal superconducting Meissner-state model with zero interior B, the exterior normal component at the boundary must therefore be zero, leaving any nonzero exterior B tangent to the surface. This conclusion is conditional on the stated zero-interior model; it does not follow by assuming all exterior field is zero. For electrostatics, normal D can instead jump by free surface charge.

A plane wave travels along positive z in a linear lossless nonmagnetic dielectric. With relative permittivity 9,

If E points along positive x, $\mathbf B=\hat{\mathbf z}\times\mathbf E/v$ points along positive y. For E0 = 200 V/m, $\mathbf E\times\mathbf H$ points along propagation. Check that material constants apply at the wave frequency.

Displacement current uses changing electric flux. Zero interior B fixes the exterior normal component, not every exterior component. Distinguish wave propagation from particle motion.

displacement current: Electric-flux time-derivative contribution in Ampere–Maxwell law.

Poynting vector: Electromagnetic energy flux per unit area and time.