RC 1 RC Circuits An RC circuit is a circuit with a resistor R and a capacitor C RC circuits are used to construct timers and filters Example 1 Very simple RC circuit a capacitor C charged to an initial voltage V0 Q0 C attached to a resistor R with a switch switch Q0 C C R Q0 Q V V0 Q0 C Close the switch at time t 0 so current I starts to flow The charged capacitor is acting like a battery it produces a voltage difference across the resistor which drives the current through the resistor At t 0 C Q I Q R I Vacross C Vacross R VC VR dQ dt I0 V0 R sign because Q is decreasing Q Q dQ IR R C C dt dQ 1 Q dt RC RC time constant has units of time is a differential equation of the form dx a x where a is a constant dt This equation says rate of charge of x x exponential solution x x 0 exp a t d x 0 ea t dx Check a x 0 ea t a x dt dt a 0 exponential growth The solution to dQ 1 Q dt RC It works a 0 exponential decay is t t Q t Q 0 exp C V0 exp RC Notice that at t 0 the formula gives Q Q0 In time Q falls by a factor of exp 1 1 e 0 37 Last update 10 8 2009 Dubson Phys1120 Notes University of Colorado RC 2 In time 2 Q falls by a factor of exp 2 1 e 1 e 0 14 Q approaches zero asymptotically and so does V and I Q t t Q t Q 0 exp Q 0 exp RC Q0 Q0 e t I V Q C V dQ 0 exp t dt R t t Example 2 More complex RC circuit Charging a capacitor with a battery switch E C VC Q C Let s use symbol E for battery voltage E short for emf because there are so many other V s in this example Before switch is closed I 0 Q 0 R VR I R Close switch at t 0 Always true that E VC VR by Kirchhoff s Voltage Law Loop Law The charge Q on the capacitor and the voltage VC Q C across the capacitor cannot change instantly since it takes time for Q to build up so At t 0 Q 0 VC 0 E VC VR VR I R I0 E R Although Q on the capacitor cannot change instantly the current I dQ dt can change instantly Current through a capacitor means dQ dt Even though there is no charge ever passing between the plates of the capacitor there is a current going into one Last update 10 8 2009 I Dubson Phys1120 Notes University of Colorado I RC 3 plate and the same current is coming out of the other plate so it is as if there is a current passing through the capacitor C E E VR VC Q C E IR I R E I dQ dt dQ Q R dt C Qualitatively as t Q VC Q C VR I VR R After a long time t RC the current decreases to zero I 0 VC E Q C E Vc Q C Analytic solution E VC t E 1 exp t RC Q t E C 1 exp t RC t Things to remember Uncharged capacitor acts like a short a wire since VC Q C 0 After a long time when the capacitor is fully charged it acts like an open circuit a break the wire We must have IC 0 eventually otherwise Q VC Last update 10 8 2009 Dubson Phys1120 Notes University of Colorado
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