맥스웰 파동, 에너지 흐름 및 경계 조건
도입됨| English | 의미 |
|---|---|
| displacement current/dɪˈspleɪsmənt ˈkʌrənt/ | 변위 전류 |
| permittivity/ˌpɜːmɪˈtɪvəti/ | A property relating electric displacement to electric field; in a linear isotropic material, D equals permittivity times E. |
| Poynting vector/ˈpɔɪntɪŋ ˈvektə/ | 포인팅 벡터 |
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.
Plates of area 0.010 m²; the gap field rises at $10^{12}$ V/(m·s). Displacement current in mA (one decimal place)? Use $\varepsilon_0 = 8.85\times10^{-12}$.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
$8.85\times10^{-12} \times 0.01 \times 10^{12}$ A.
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.
Wave speed in a nonmagnetic medium with $\varepsilon_r = 4$, in units of $10^8$ m/s?
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
In a nonmagnetic medium $v = c/\sqrt{\varepsilon_r} = c/2 = 1.5\times10^8$ m/s.
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 $-z$.
Travelling $+z$ with $\mathbf E \parallel \hat x + \hat y$: $\hat z \times \hat x = \hat y$, $\hat z \times \hat y = -\hat x$, so $\mathbf B \parallel -\hat x + \hat y$, and $\mathbf E \times \mathbf B$ points along $+z$.
A wave travels along $+x$ with $\mathbf E$ along $+y$. $\mathbf B$ points along
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
$\hat x \times \hat y = \hat z$.
$\mathbf E = E_0\cos(kz + \omega t)\,\hat x$ travels toward
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Constant phase needs z decreasing as t grows.
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.
Continuous or can jump?
Recall Maxwell boundary conditions at an interface.
Outside an ideal superconductor the magnetic field must be zero everywhere at its surface.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
Only the normal component must vanish.
Recap
- Displacement current $\varepsilon_0\,d\Phi_E/dt$ completes Ampère's law.
- $v = c/\sqrt{\varepsilon_r\mu_r}$; $\mathbf B \propto \hat{\mathbf n} \times \mathbf E$; $\mathbf S$ along travel.
- Normal $B$ and tangential $E$ are continuous; the others jump by surface sources.