作者SmArTyau (SmArT)
看板Physics
標題Re: [題目] drift velocity 的微分方程
時間Sat May 15 01:07:43 2010
※ 引述《ian60702 (pacific hero)》之銘言：
: [領域] 電磁學 (題目相關領域)
: [來源] 習題 (課本習題、考古題、參考書...)
: [題目]
: (摘要)
: Show that the steady-state solution to the following eqation is given by
: τe
: v = ────────── Ecos(wt-tan^-1 (wτ))
: m [(1+(wr)^2]^(1/2)
^ 應該是τ
: m
: mv'+ ─ v = eEcoswt
: τ
: 設v(0)=0
: [瓶頸] (寫寫自己的想法，方便大家為你解答)
: 應該是數學的問題= =''
: 不是只是很單純的用未定係數法去算嗎
: 怎麼答案會這麼奇怪..
: 懇請高手解惑
(Dt+1/τ)v = (eE/m) cos(wt)
vp= (eE/m) cos(wt) / (Dt+1/τ)
= [ (Dt-1/τ)/(Dt^2-1/τ^2) ] (eE/m) cos(wt)
= (Dt-1/τ)/(-w^2-1/τ^2) (eE/m) cos(wt)
= [(eE/m)/(-w^2-1/τ^2)] * [ (-w)sinwt -(1/τ)coswt]
= [(eE/m)/(-w^2-1/τ^2)] * {-sqrt[(w)^2+(1/τ)^2]*cos(wt-tan^-1(wτ))}
= [(eE/m)/sqrt(w^2+1/τ^2)]*cos(wt-tan^-1(wτ))
= {(eEτ/m)/sqrt[(wτ)^2+1]}*cos(wt-tan^-1(wτ))
vh= A exp[-t/τ]
v=vh+vp
v(0)=A+[(eE/m)/(-w^2-1/τ^2)]*(-1/τ)
A= [-(eE/mτ)/(w^2+1/τ^2)]
不過vh是exponential decay 不算是STEADY STATE的部分
故取vp的部分即可
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◆ From: 114.32.29.237
※ 編輯: SmArTyau 來自: 114.32.29.237 (05/15 01:17)
1F：推 ian60702:高手! 05/15 22:59