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# lecture16 - RC Circuits nts e circuits in which thecurre...

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RC Circuits RC Circuits - circuits in which the currents vary in time - rate of charging a cap depends on C and R of circuit - differential equations

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Discharging a Capacitor Discharging a Capacitor ( - sign because q decreases for I > 0 That is, current in circuit equals the decrease of charge on the capacitor) C R q -q I Given: R, C, q o (initial charge) Find: q(t) and I(t) when switch is closed dt dq I - = 2) 1) 0 = - IR C q (Kirchhoff’s Loop Rule)
C R q -q I RC q dt dq - = where: q = q(t) q(0) = q o This is a differential equation for the function q(t), subject to the initial condition q(0) = q 0 . We arelooking for a function which is proportional to its own first derivative (since dq/dt ~ -q). Combine 1) and 2) to get:

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RC is called the“timeconstant ” or “characteristic time” of the circuit. Units: 1 x 1 F = 1 second (show this!) Ω Write τ (“tau”) = RC, then: - = τ t o e q t q ) ( (discharging) RC t ο e q q(t) - = Solution:
Discharging Discharging q q o τ τ τ 2 3 t τ t o e q t q - = ) ( RC τ t = , q 0.37 q o = (q o /e) t = 2 , q 0.14 q o = (q o

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