Simple Nature - Light and Matter

The RC circuitWe first analyze the RC circuit, o. In reality one would haveto “kick” the circuit, for example by briefly inserting a battery, inorder to get any interesting behavior. We start with Ohm’s law andthe equation for the voltage across a capacitor:V R = IRV C = q/Co / An RC circuit.The loop rule tells usV R + V C = 0 ,and combining the three equations results in a relationship betweenq and I:I = − 1RC qThe negative sign tells us that the current tends to reduce the chargeon the capacitor, i.e., to discharge it. It makes sense that the currentis proportional to q: if q is large, then the attractive forces betweenthe +q and −q charges on the plates of the capacitor are large,and charges will flow more quickly through the resistor in order toreunite. If there was zero charge on the capacitor plates, there wouldbe no reason for current to flow. Since amperes, the unit of current,are the same as coulombs per second, it appears that the quantityRC must have units of seconds, and you can check for yourself thatthis is correct. RC is therefore referred to as the time constant ofthe circuit.How exactly do I and q vary with time? Rewriting I as dq/ dt,we havedqdt = − 1RC q .We need a function q(t) whose derivative equals itself, but multipliedby a negative constant. A function of the form ae t , where e =2.718... is the base of natural logarithms, is the only one that has itsderivative equal to itself, and ae bt has its derivative equal to itselfmultiplied by b. Thus our solution isThe RL circuitq = q o exp(− t )RCThe RL circuit, q, can be attacked by similar methods, and itcan easily be shown that it givesI = I o exp(− R )L t .The RL time constant equals L/R..p / Over a time interval RC,the charge on the capacitor isreduced by a factor of e.q / An RL circuit.Section 10.5 LRC Circuits 601