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lecture21 - C 19 LC circuits Kirchhoff's loop rule vC = ind...

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2 2 dt q d L dt di L ind - = - = ε 19. LC circuits C q v / = const C Q Li C q U U U B E = = + = + = 2 2 2 2 2 2 L C ε Conservation of energy: Kirchhoff’s loop rule: 0 2 2 2 2 = + - = = LC q dt q d dt q d L C q v ind C ε 0 2 2 2 = + q dt q d ϖ LC 1 = ϖ Solution: ( 29 ( 29 ( 29 ( 29 φ ϖ φ ϖ ϖ φ ϖ + = + - = + = t C Q v t Q i t Q q C cos / sin cos T f π π ϖ 2 2 = = Important: current and potential difference between the capacitor’s plates are not in phase (shifted by 90 )
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2 2 dt q d L dt di L ind - = - = ε 20. LRC circuits C q v C / = L C ε Kirchhoff’s loop rule: + = iR v C ind ε 0 2 2 0 2 2 = + + q dt dq dt q d ϖ λ LC 1 0 = ϖ Solution: ( 29 ( 29 [ ] ( 29 ( 29 iR v t e C A C q v t t Ae dt dq i R t C t = + = = + + + - = = - - φ ϖ φ ϖ ϖ φ ϖ λ λ λ ' cos / / ' sin ' ' cos / R 0 2 2 2 2 = + + + = - LC q dt dq L R dt q d dt dq R C q dt q d L L R 2 = λ ( 29 φ ϖ λ + = - t Ae q t ' cos 2 2 0 ' λ ϖ ϖ - =
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0 2 2 0 2 2 = + + q dt dq dt q d ϖ λ Solution of the differential equation: ( 29 ( 29 g e g e g e dt q d q g e g e dt dq q t g e t q t t t t t t λ λ λ λ λ λ λ λ λ - - - - - - + - = - = 2 2 2 2 , ( 29 = + - + + - 0 2 2 2 0 2 g g g g g g ϖ λ λ
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