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Elektromagnit induksiya qonuni

Elektromagnit induksiya qonuni yoki Faradayning induksiya qonuni

Modomiki zarralarning kinetik energiyasi oʻzgarishi ularning tezligi bilan bogʻlangan ekan, biz darhol harakat tezligi va elektromagnit maydon kuchlanganliklari orasidagi bogʻlanishni ifodalovchi quyidagi formulaga murojaat qilamiz:

rotH=4πρuc+1c∂E∂t{\displaystyle \textrm{rot}\textbf{H}=4\pi\frac{\rho\textbf{u}}{c}+\frac{1}{c}\dfrac{\partial \textbf{E}}{\partial t}}

Mazkur tenglamadan tok zichligini aniqlaymiz:

ρuc4πrotH−14π∂E∂t{\displaystyle \rho\textbf{u}\frac{c}{4\pi}\textrm{rot}\textbf{H}-\frac{1}{4\pi}\dfrac{\partial\textbf{E}}{\partial t}};(1)

Bu tenglamaning ikki tomonini E{\displaystyle \textbf{E}} vektorga skalyar koʻpaytirib chiqamiz:

(ρuE)=c4π(ErotH)−14π(E∂E∂t);(2){\displaystyle (\rho\textbf{uE})=\frac{c}{4\pi}\left(\textbf{E}\textrm{rot}\textbf{H}\right)-\frac{1}{4\pi}\left(\textbf{E}\dfrac{\partial \textbf{E}}{\partial t}\right) ;(2)}

Har qanday a{\displaystyle \textbf{a}} vektor uchun quyidagini yozish mumkin:

∂a2∂t=∂∂t(aa)=(a∂a∂t)+(∂a∂ta)=2(a∂a∂t){\displaystyle \dfrac{\partial a^{2}}{\partial t}=\dfrac{\partial}{\partial t}(\textbf{aa})=\left(\textbf{a}\dfrac{\partial a}{\partial t}\right)+\left(\dfrac{\partial a}{\partial t}\textbf{a}\right)=2\left(\textbf{a}\dfrac{\partial a}{\partial t}\right)}

demak,

(E∂E∂t)=12∂E2∂t,{\displaystyle \left(\textbf{E}\dfrac{\partial \textbf{E}}{\partial t}\right)=\frac{1}{2}\dfrac{\partial \textbf{E}^{2}}{\partial t},}
(H∂H∂t)=12∂H2∂t{\displaystyle \left(\textbf{H}\dfrac{\partial\textbf{H}}{\partial t}\right)=\frac{1}{2}\dfrac{\partial\textbf{H}^{2}}{\partial t}}

Maydon nazariyasidan maʼlumki, vektor koʻpaytmaning divergensiyasi uchun div[EH]=(HrotE)−(ErotH){\displaystyle \textrm{div} [\textbf{EH}]=(\textbf{H}\ \textrm{rot}\ \textbf{E})-(\textbf{E} \textrm{rot} \textbf{H})}, demak, (ErotH)=(HrotE)−div[EH]{\displaystyle (\textbf{E} \textrm{rot} \textbf{H})=(\textbf{H}\ \textrm{rot} \textbf{E})-\textrm{div}\ [\textbf{EH}]}, demak, (ErotH)=(HrotE)−div[EH]{\displaystyle \textbf{E}\ \textrm{rot}\textbf{H})=(\textbf{H}\ \textrm{rot}\textbf{E})-\textrm{div}[\textbf{EH}]}. Endi (2)-tenglamani quyidagicha yozib olaylik:

(ρuE)=c4π(HrotE)−c4πdiv[EH]−18π∂∂tE2+14π(H∂H∂t)−18π∂∂tH2{\displaystyle (\rho\textbf{uE})=\frac{c}{4\pi}(\textbf{H}\ \textrm{rot}\textbf{E})-\frac{c}{4\pi}\textrm{div}[\textbf{EH}]-\frac{1}{8\pi}\dfrac{\partial}{\partial t}E^{2}+\frac{1}{4\pi}\left(\textbf{H}\dfrac{\partial\textbf{H}}{\partial t}\right) -\frac{1}{8\pi}\dfrac{\partial}{\partial t}H^{2}}

yoki

(ρuE)=14π(H,crotE+∂H∂t)−∂∂t[18πE2+18πH2]−divc4π[EH]{\displaystyle (\rho\textbf{uE})=\frac{1}{4\pi} \left(\textbf{H},c\ \textrm{rot}\textbf{E}+ \dfrac{\partial\textbf{H}} {\partial t}\right) -\dfrac{\partial}{\partial t}\left[ \frac{1}{8\pi}E^{2}+ \frac{1}{8\pi}H^{2} \right] -\textrm{div} \frac{c}{4\pi}\left[\textbf{EH}\right]}

Mazkur ifodani zaryadlar va elektromagnit maydonlar joylashgan ixtiyoriy hajm boʻyicha integrallab chiqamiz: ∫(ρuE)dV=14π∫(H,crotE+∂H∂t)+∮(c4π[EH],dσ)+∫(ρuE)dV=14π∫(H,crotE+∂H∂t)dV{\displaystyle \int (\rho\textbf{uE})dV= \frac{1}{4\pi}\int \left( \textbf{H}, c\ \textrm{rot}\textbf{E}+ \dfrac{\partial \textbf{H}}{\partial t} \right) +\oint\left( \frac{c}{4\pi} [\textbf{EH}],d\sigma \right) +\int(\rho\textbf{uE})dV=\frac{1}{4\pi}\int\left( \textbf{H},c\ \textrm{rot}\textbf{E}+\dfrac{\partial \textbf{H}}{\partial t} \right) dV}

Integrallash hajmining oʻzgarmasligi tufayli va Gauss-Ostrogradskiy formulasi asosida yuqoridagi ifoda shunday yoziladi:

ddt∫(E28π+H28π)dV+∮(c4π[E,H],dσ)+∫(ρuE)dV=14π∫(H,crotE+∂H∂t)dV;(3){\displaystyle \frac{d}{dt}\int\left(\frac{E^{2}}{8\pi}+\frac{H^{2}}{8\pi}\right)dV+ \oint\left(\frac{c}{4\pi}\ [\textbf{E,H}],d\sigma\right) +\int(\rho \textbf{uE})dV=\frac{1}{4\pi} \int\left(\textbf{H}, c\ \textrm{rot}\ \textbf{E}+\dfrac{\partial \textbf{H}}{\partial t}\right)dV ;(3)}

Shu topilgan natija gʻoyat katta ahamiyatga ega. Agar soʻnggi tenglikning chap tomonidagi birinchi had hajmdagi elektromagnit maydon energiyasining vaqt birligidagi oʻzgarishi, ikkinchi had hajm chegaralangan yopiq sirtdan vaqt birligida oʻtayotgan elektromagnit energiya oqimi, uchinchi had elektromagnit maydon taʼsiri natijasida hajmdagi zarralar kinetik energiyasining vaqt birligidagi oʻzgarishi deb qabul qilinsa, (1) va (3) ga asosan energiya saqlanish qonuni quyidagi shaklni oladi:

ddt∫(E28π+H28π)dV+∮(c4π[EH],dσ)+∫(ρuE)dV=0;(4){\displaystyle \dfrac{d}{dt}\int\left(\frac{E^{2}}{8\pi}+\frac{H^{2}}{8\pi}\right)dV + \oint\left(\frac{c}{4\pi}[\textbf{E H}], d\sigma \right)+ \int(\rho \textbf{u E}) dV = 0 ;(4)}

Bu muhim formula elektrodinamikada energiya saqlanish qonunini ifodalaydi.

Energiya saqlanish qonunining (4) ifodasiga muvofiq, (3) dan koʻramizki,

∫(H,crotE+∂H∂t)dV=0{\displaystyle \int\left(\textbf{H}, c\ \textrm{rot} \textbf{E} + \dfrac{\partial H}{\partial t} \right)dV = 0}

Integrallash hajmi ixtiyoriy qilib olingan edi, demak,

(H,crotE+∂H∂t)=0{\displaystyle \left(\textbf{H}, c\ \textrm{rot} \textbf{E} + \dfrac{\partial H}{\partial t} \right) = 0}

Bu natija har qanday E va H vektorlar uchun toʻgʻridir, demak

crotE+∂H∂t=0{\displaystyle c\ \textrm{rot}\textbf{E} + \frac{\partial H}{\partial t} = 0}

yoki

rotE=−1c∂H∂t{\displaystyle \textrm{rot} \textbf{E} = -\frac{1}{c}\frac{\partial H}{\partial t}}

Bu ifoda elektrodinamikaning asosiy differensial tenglamalaridan biridir. Mazkur differensial tenglama magnit maydon oʻzgarishi natijasida uyurmaviy elektr maydon paydo boʻlishi qonunini ifodalaydi. Bu qonun elektromagnit induksiya qonuni deb ham ataladi.

Yana qarang

Adabiyotlar

  • R.X.Mallin, Klassik elektrodinamika, Oʻqituvchi, T., 1974

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