经典场论微观视角：导体表面电荷与斜入射电磁波E、H分量作用问询

Let’s cut straight to the microscopic grit here—forget reciting those boundary condition equations for a second. I’ll walk you through exactly what’s happening with the free electrons and surface charges when an obliquely incident EM wave hits a conductor:

1. The starting state: Free electrons in equilibrium #

Conductors (think copper, aluminum) have a sea of loosely bound, freely mobile electrons. When things are calm, these electrons zip around randomly, so there’s no net macroscopic charge or current on the surface. That all changes the second the EM wave arrives.

2. What the incident wave brings to the party #

An obliquely incident plane wave has two key electric field components relative to the conductor surface:

• A tangential E component (parallel to the surface)
• A normal E component (perpendicular to the surface)
  It also carries a magnetic field (H) with its own tangential/normal components—remember, E and H are always perpendicular in propagating EM waves, and their magnitudes are linked by the impedance of free space.

3. Tangential E kicks off surface currents #

The tangential E field exerts a direct force on free electrons: F = -eE_{\text{tan}} (the negative sign accounts for electrons being negatively charged). This shoves electrons along the surface, creating a surface current density (K).

Now, this moving current generates its own magnetic field. By Lenz’s law, this induced field pushes back against the incident wave’s tangential H component inside the conductor. For an ideal conductor (infinite conductivity), this cancellation happens instantaneously—no H field survives inside the conductor. In real conductors, there’s a tiny "skin depth" where the field decays exponentially, and the current dissipates energy as heat (Joule loss) because electrons collide with the conductor’s atomic lattice.

Also, don’t overlook the Lorentz force from the incident H field: moving electrons (the current) experience F = -e(v × B) (where B = μ₀H). This force acts perpendicular to both the electron’s motion and the H field—so it pulls or pushes electrons toward/away from the surface, tying directly into charge accumulation.

4. Normal E piles up surface charges #

The normal E field tugs or shoves electrons toward the surface (or away, depending on the field’s direction). For example, if E_{\text{norm}} points into the conductor, electrons (negative charge) get pushed toward the surface, piling up as a surface charge density (σ).

This accumulated charge generates its own normal electric field, which opposes the incident wave’s normal E component inside the conductor. Again, ideal conductors achieve perfect cancellation—no normal E field exists inside. Real conductors have a tiny internal field, but it’s negligible compared to the incident wave.

5. The synchronized dance of charges and currents #

In an obliquely incident wave, the tangential E and H components are in phase (for a plane wave in vacuum). That means the surface current and surface charge change in lockstep: as the tangential E grows, the current increases, and the Lorentz force modulates how much charge piles up on the surface. It’s a dynamic equilibrium—charges and currents adjust instantly (in ideal conductors) to snuff out any internal fields.

6. How this connects to the boundary conditions we know #

When we zoom back to the macroscopic view, the boundary conditions are just shorthand for these microscopic dynamics:

• The surface charge σ links to the normal E discontinuity: σ = ε₀(E_{\text{norm, outside}} - E_{\text{norm, inside}})
• The surface current K links to the tangential H discontinuity: K = \hat{n} × (H_{\text{outside}} - H_{\text{inside}})

These equations don’t come out of thin air—they’re just the macroscopic summary of electrons moving and charging up the surface to cancel internal fields.

内容的提问来源于stack exchange，提问作者2physics