Maxwell waves, energy flow and boundary conditions
| English | Português |
|---|---|
| displacement current | displacement current |
| Poynting vector | Poynting vector |
A decision before an answer
- A charged capacitor can change its electric field across a gap even though no conduction current crosses the gap.
- Your goal: Use displacement current and material wave-speed relations.
Read the relationship
- 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.
- Determine wave-field directions and radiated energy flow.
For a +z plane wave whose electric field is along +x, magnetic field points:
z×x=y, ensuring E×B points +z.
Use the defining rule
- In a homogeneous, linear, lossless dielectric 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.
- Apply Maxwell boundary conditions without confusing field components.
If interior B is zero at an ideal Meissner boundary, a nonzero outside B must be:
Normal B is continuous and therefore zero; a tangential component can remain.
Check the conditions
- 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 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.
- Apply Maxwell boundary conditions without confusing field components.
A nonmagnetic medium with εr=9 has wave speed c/3=1.0×10⁸ m/s. For a +z wave with E proportional to x+y, magnetic direction is −x+y and E×B points +z. At a plane Meissner boundary with zero interior B, an exterior field may be tangential but cannot have a nonzero normal component.
In a nonmagnetic lossless dielectric with εr=16, wave speed is ____ times c (decimal).
v/c=1/sqrt(16)=1/4.
Apply the task format
- 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.
- Apply Maxwell boundary conditions without confusing field components.
Displacement current uses changing electric flux. Zero interior B fixes the exterior normal component, not every exterior component. Distinguish wave propagation from particle motion.
Which answer fits this case?
Use displacement current and material wave-speed relations
For capacitor charging, displacement current depends on the rate of change of electric flux rather than magnetic flux.
The gap term is ε0dΦE/dt in vacuum.
Keep the distinctions
- displacement current 位移电流 — Electric-flux time-derivative contribution in Ampere–Maxwell law.
- Poynting vector 坡印廷矢量 — Electromagnetic energy flux per unit area and time.
- Use displacement current and material wave-speed relations.
- Determine wave-field directions and radiated energy flow.
- Apply Maxwell boundary conditions without confusing field components.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.