The Rc Circuit

%23+The+RC+Circuit%0A%0AThe+capacitor+holds+a+charge.+The+resistor+resists+flow.+Put+them+together+and+the+story+begins.%0A%0A%23%23+The+Charge%0A%0AAt+time+zero%2C+the+capacitor+is+full.+There+is+energy+stored+in+its+plates%E2%80%94positive+charge+here%2C+negative+charge+there%2C+separated+by+a+gap+that+nothing+can+cross.+The+voltage+across+it+is+%24V_0%24.+The+energy+sitting+in+that+gap+is+%24E+%3D+%5Cfrac%7B1%7D%7B2%7DCV_0%5E2%24%2C+waiting+to+be+released.%0A%0AThe+resistor+sits+in+series%2C+a+narrow+bridge+between+the+two+capacitor+plates.+Charges+on+the+positive+plate+want+to+leave.+They+feel+the+electric+field+pulling+them+away+from+their+like-charged+neighbors.+They+want+to+flow+through+the+resistor%2C+neutralize+the+electrons+on+the+other+plate%2C+and+return+to+equilibrium.%0A%0AThe+resistor+lets+them%2C+but+slowly.+Every+coulomb+that+flows+through+the+resistor+pays+a+price%E2%80%94heat%2C+dissipation%2C+the+irreversible+loss+of+organized+charge+into+disorganized+thermal+motion.+The+resistor+is+the+bottleneck.+It+controls+the+rate.%0A%0A%23%23+The+Discharge%0A%0AThe+circuit+is+closed.+A+switch+throws.+The+story+is+written+by+differential+equations.%0A%0AThe+current+at+any+moment+is+determined+by+the+voltage+across+the+resistor%2C+which+is+the+same+as+the+voltage+across+the+capacitor%3A+%24I%28t%29+%3D+%5Cfrac%7BV_C%28t%29%7D%7BR%7D%24.+But+the+capacitor+voltage+is+falling+because+charge+is+leaving%3A+%24I%28t%29+%3D+-C%5Cfrac%7BdV_C%7D%7Bdt%7D%24.+The+minus+sign+matters%E2%80%94it+says+voltage+decreases+as+charge+flows+out.%0A%0AEquating+these+gives+the+governing+equation%3A+%24RC%5Cfrac%7BdV_C%7D%7Bdt%7D+%2B+V_C+%3D+0%24.+The+solution+is+inevitable%3A%0A%0A%24V_C%28t%29+%3D+V_0+e%5E%7B-t%2FRC%7D%24%0A%0AThe+voltage+decays+exponentially.+The+time+constant+%24%5Ctau+%3D+RC%24+is+the+characteristic+timescale.+After+one+time+constant%2C+the+voltage+has+dropped+to+%241%2Fe+%5Capprox+36.8%5C%25%24+of+its+initial+value.+After+five+time+constants%2C+it+is+below+1%25.+For+practical+purposes%2C+the+capacitor+is+empty.%0A%0A%23%23+What+the+Time+Constant+Means%0A%0A%24R%24+in+ohms%2C+%24C%24+in+farads%2C+%24%5Ctau%24+in+seconds.+The+units+combine+naturally%E2%80%94resistance+times+capacitance+is+a+time.+Think+of+what+each+represents.+A+large+resistor+means+a+narrow+bottleneck%E2%80%94charge+flows+slowly.+A+large+capacitor+means+a+big+reservoir%E2%80%94there+is+more+charge+to+lose.+Both+make+the+discharge+slower.+A+large+resistor+and+a+small+capacitor%3F+The+bottleneck+is+severe+but+the+reservoir+is+tiny.+The+outcome+depends+on+the+product+%24RC%24%2C+not+on+either+alone.%0A%0AThis+is+why+time+constants+appear+everywhere+in+electronics.+Every+circuit+with+resistance+and+capacitance+has+one.+A+filter+that+blocks+high+frequencies.+A+delay+that+turns+on+an+LED+half+a+second+after+a+button+press.+A+smoothing+circuit+that+reduces+ripple+in+a+power+supply.+All+the+same+physics%2C+different+numbers.%0A%0A%23%23+The+Charge+Story%0A%0AReverse+the+story%3A+connect+an+uncharged+capacitor+in+series+with+a+resistor+to+a+voltage+source.+The+current+starts+high%E2%80%94instantly%2C+the+capacitor+looks+like+a+short+circuit%E2%80%94and+decays+as+the+capacitor+charges.+%24I%28t%29+%3D+%5Cfrac%7BV%7D%7BR%7De%5E%7B-t%2FRC%7D%24.+The+capacitor+voltage+rises+as+%24V_C%28t%29+%3D+V%281+-+e%5E%7B-t%2FRC%7D%29%24%2C+asymptotically+approaching+the+source+voltage.%0A%0ACharge+and+discharge+are+mirror+images.+Both+governed+by+the+same+time+constant.+Both+exponential.+Both+inevitable.%0A%0A%23%23+The+Energy%0A%0AHere+is+the+thing+nobody+tells+you%3A+exactly+half+the+energy+supplied+by+the+source+is+stored+in+the+capacitor+during+charging.+The+other+half+is+dissipated+as+heat+in+the+resistor.+This+is+true+regardless+of+resistance+value.+Even+with+zero+resistance%E2%80%94where+intuition+suggests+no+dissipation+should+occur%E2%80%94half+the+energy+is+lost.+%28In+the+zero-resistance+limit%2C+the+circuit+oscillates+and+the+energy+loss+mechanism+changes%2C+but+the+result+holds+for+any+nonzero+%24R%24.%29%0A%0AThe+resistor+does+not+merely+slow+things+down.+It+destroys+energy.+Irreversibly.+The+capacitor+can+give+it+back%2C+but+only+as+electrical+energy.+The+heat+in+the+resistor+is+gone%E2%80%94scattered+into+the+thermal+motion+of+trillions+of+atoms.