Maxwell's equations
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| charge conservation/tʃɑːdʒ ˌkɒnsəˈveɪʃn/ | charge conservation |
The missing ampere
- Charge conservation forced Maxwell's correction: without $\varepsilon_{0}\partial E/\partial t$ the Ampère law contradicts $\nabla \cdot J = -\partial\rho/\partial t$. With it, the four equations form one loop.
A capacitor charging at 1 A: between the plates $\varepsilon_{0}A\,\partial E/\partial t = dQ/dt = 1$ A ✓ — displacement equals conduction, numerically.
{exercise:uni-ee0741-capacitor}
Between the plates of a charging capacitor, an Amperian loop reads —
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ε0·A·∂E/∂t = dQ/dt exactly — the loop integral cannot depend on which surface you span it with.
Which equation carries the minus sign?
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The minus sign in Faraday (Lenz) is what makes waves decay forward in time rather than blow up backward.
Light falls out
- Take the curl of Faraday, substitute Ampère: in vacuum $\nabla^{2}E = \mu_{0}\varepsilon_{0}\,\partial^{2}E/\partial t^{2}$ — waves at $c = 1/\sqrt{\mu_{0}\varepsilon_{0}} = 3 \times 10^{8}$ m/s from two bench constants.
{exercise:uni-ee0742-speed}
The wave speed 1/√(μ0ε0) evaluates to —
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Two bench-measured constants give 1/√(4π×10⁻⁷ × 8.854×10⁻¹²) = 3.0×10⁸ m/s — the speed light was hiding in.