Poyntiny's Theorem
We = ε/2 (E‧E)
Wm = 1/2μ(B‧B)
Uem = 1/2(ε(E‧E) + 1/μ(B‧B))
Uem is called energy density
Accroding to the Lorentz force law
F‧dl = q(E + V ×B)‧Vdt
magnetic forces do no work
F‧dl = qE‧Vdt
q = ρdτ
F‧dl = E‧ρVdτdt
dW/dt = ∫(E‧J)dτ
Accroding to the Ampere's law with Maxwell's equation correction
▽ ×B = μJ+εμ(dE/dt)
J = 1/μ(▽ ×B) － ε(dE/dt)
=> dW/dt = ∫(1/μE‧(▽ ×B) － εE‧(dE/dt))dτ
E‧(▽ ×B) = B‧(▽ ×E) － ▽‧(E ×B)
Accroding to the Faraday's law
▽ ×E = －(dB/dt)
=> E‧(▽ ×B) = －B‧(dB/dt) － ▽‧(E ×B)
dW/dt = ∫(－1/μB‧(dB/dt) － εE‧(dE/dt))dτ － 1/μ▽‧(E ×B))dτ
B‧(dB/dt) = 1/2(d(B‧B)/dt) E‧(dE/dt) = 1/2(d(E‧E)/dt)
dW/dt = (d/dt)(－1/2)∫(ε(E‧E) + 1/μ(B‧B))dτ － 1/μ∮(E ×B)‧da
Poynting's vector S = 1/μ(E ×B)
The first integral on the right is the tatle energy in the electromagnetic
field.While the sencond integral is the energy flow on boundary surface of
the volume V.
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I don't teach easy electromagnetics,but I teach electromagnetics easily.
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◆ From: 218.161.8.174
※ 編輯: Hydrisk 來自: 218.161.8.174 (04/16 14:07)