Sóng Maxwell, dòng năng lượng và điều kiện biên

A magnetic field across an empty gap

• While a capacitor charges, no conduction current crosses the gap — yet a magnetic field circles it, as if the current continued.
• GRE electromagnetism tests displacement current 位移电流, waves in media, field directions and boundary conditions.

Displacement current

• $\oint \mathbf B \cdot d\mathbf l = \mu_0\left(I + \varepsilon_0\dfrac{d\Phi_E}{dt}\right)$. Between plates, $I_d = \varepsilon_0A\,dE/dt$ equals the wire current.
• It is a rate of change of electric flux, needed so that any surface on the same loop gives the same answer.

Waves in a medium

• $v = 1/\sqrt{\mu\varepsilon}$, $n = c/v = \sqrt{\mu_r\varepsilon_r}$. Nonmagnetic: $v = c/\sqrt{\varepsilon_r}$, so $\varepsilon_r = 9$ gives $v = 1.0\times10^8$ m/s. Then $E/B = v$.
• Use the permittivity 介电常数 at the wave's frequency; the static value may not apply at optical frequencies.

Directions

• $\mathbf B = \hat{\mathbf n} \times \mathbf E/v$; the Poynting vector 坡印廷矢量 $\mathbf S = \mathbf E \times \mathbf H$ points along energy flow. $kz - \omega t$ travels toward $+z$; $kz + \omega t$ toward $-z$.

Boundary conditions

• $\nabla \cdot \mathbf B = 0$: normal $B$ is continuous. Tangential $E$ is continuous. Normal $D$ jumps by free surface charge; tangential $H$ jumps by surface current.
• Ideal superconductor (Meissner, $B = 0$ inside): outside, $B_\perp = 0$ at the surface — the field may run along it.

Recap