RC Circuits (Charging) ! ! _ + Qo = 0 RC Circuits a b R C a R +q -q b C c " # v ab # v bc = 0 " # iR # Cq = 0 " q i= # R ! 1 i i c ! Current, Resistance, and emf _ + RC Current, Resistance, and emf 2 ! RC Circuits (Charging) " i= # R RC Circuits (Charging) q RC q t dq dt =$" q " C RC 0 0 $ # $ q " C# ' t ln& )=" % "C# ( RC q " C# =e "C# " ! ! dq q = # dt R RC ! dq dt =" q"C RC # "t ! RC q ! t dq dt =$" $ q " C RC 0 0 ( # Current, Resistance, and emf q(t ) = Q f 1 " e ! 3 "t RC ) where Q f = C # Current, Resistance, and emf ! $ 4 ! RC Circuits (Charging) RC Circuits (Charging) q The charge on the capacitor varies according to: Qf ( q(t ) = Q f 1 " e "t RC ( ) q(t ) = Q f 1 " e RC ) t The current at any time is given by: #t "t ! dq i= = e RC = Io e RC dt R " i ! Io RC is called the time constant (" ) and is the time it takes the capacitor to ! become 63.2% charged. ! ! Current, Resistance, and emf "t i = Io e "t RC Io /e t RC 5 Current, Resistance, and emf ! 6 RC Circuits (Charging) RC Circuits (Discharging) q i Qf i=0 a R i +Qo -Qo b Increasing RC C c a iR + t i= ! Current, Resistance, and emf 7 R +q -q b C c q =0 C dq q =" dt RC Current, Resistance, and emf 8 ! RC Circuits (Discharging) RC Circuits (Discharging) dq q =" dt RC The charge on the capacitor varies according to: q q = Qoe dq t dt ="# Qo q 0 RC " ! q = Qoe "t RC The current at any time is given by: ! " q% t ln$ ' = ( RC # Qo & ! "t i= RC "t Q "t dq = " o e RC = Io e RC RC dt ! Current, Resistance, and emf 9 Current, Resistance, and emf ! ! ! RC Circuits (Discharging) 10 ! RC Circuits (Discharging) q q = Qoe "t q Qo RC Qo Qo /e t RC i ! Increasing RC Io i = Io e "t RC t Io /e t RC Current, Resistance, and emf ! 11 Current, Resistance, and emf 12 Example 1 C S C S + + E# E# R An ideal battery with an emf E = 100 V is connected to a resistor with resistance R = 100 ! and an initially uncharged capacitor with capacitance C = 1 µF. The circuit is completed when switch S is closed at time t = 0. a.) Find the current through and voltage across each device immediately after the switch is closed. b.) A long time after the switch is closed, find the charge on the capacitor and the voltage across the resistor. c.) What is the charge on the capacitor after 0.2 ms? d.) Find the total energy dissipated in the resistor. Current, Resistance, and emf 13 C S R1 R2 Example 2 An ideal battery with an emf E = 12 V, two resistors with resistances R1 = 4 ! and R2 = 6 !, and an initially uncharged capacitor with capacitance C = 6 µF. The circuit is completed when switch S is closed at time t = 0. a.) At time t = 2", what is the potential difference across the capacitor? b.) At time t = 2", what are the potential differences across the two resistors? Do those potential differences increase, decrease, or remain the same while the capacitor is being Current, Resistance, and emf 14 charged? Example 3 C S + E# + E# R1 R2 Example 2 (continued) c.) A long time after the switch has been closed the switch is opened. i.) Write an equation for the charge on the capacitor after the switch is opened. ii.) Write an equation for the voltage on each resistor after the switch is opened. Current, Resistance, and emf Current, Resistance, and emf (continued) E# + E# C C R1 S2 16 S1 Example 4 + R3 In the circuit above E = 1000 V, C = 10 µF, and R1 = R2 = R3 = 1 M!. The capacitor is completely uncharged when switch S is closed. a.) Determine the current through each resistor at t = 0 and t = $. b.) Draw qualitatively a graph of the potential difference V2 across R2 for t = 0 to t = $. c.) What are the numerical values of V2 at t = 0 and t = $? 15 S1 Example 4 R2 R1 R1 R2 S2 R2 At t = 0 the capacitor is uncharged and switch S1 is closed. a.) Write a differential equation that can be solved to obtain the charge on the capacitor as a function of time. After switch S1 has been closed for a long time, switch S2 gets closed at a new time t = 0. b.) Solve the differential equation in part (a) to determine the charge on the capacitor as a function of time t. d.) Sketch graphs of the current I1 in R1 and of the current I2 in R2 versus time, beginning when switch S2 is closed at new time t = 0. Clearly label which graph is I1 and which is I2. Given that E = 12 V, C = 0.06 F, and R1 = R2 = 5000 ! c.) Determine the time at which the capacitor has a voltage 4.0 V across it. Current, Resistance, and emf 17 Current, Resistance, and emf 18