RC (Resistor-Capacitor) Circuits. AP Physics C

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1 (Rsisr-Capacir Circuis AP Physics C

2 Circui Iniial Cndiins An circui is n whr yu hav a capacir and rsisr in h sam circui. Supps w hav h fllwing circui: Iniially, h capacir is UNCHARGED ( = 0 and h currn hrugh h rsisr is zr. A swich (in rd hn clss h circui by mving upwards. Th usin is: Wha happns h currn and vlag acrss h rsisr and capacir as h capacir bgins charg as a funcin f im? Which pah d yu hink i aks? V C Tim(s

3 Vlag Acrss h Rsisr - Iniially V Rsisr If w assum h bary has NO inrnal rsisanc, h vlag acrss h rsisr will b h EMF. (sc Afr a vry lng im, V cap =, as a rsul h pnial diffrnc bwn hs w pins will b ZERO. Thrfr, hr will b NO vlag drp acrss h rsisr afr h capacir chargs. N: This is whil h capacir is CHARGING.

4 Currn Acrss h Rsisr - Iniially I max =/R (sc Sinc h vlag drp acrss h rsisr dcrass as h capacir chargs, h currn acrss h rsisr will rach ZERO afr a vry lng im. N: This is whil h capacir is CHARGING.

5 Vlag Acrss h Capacir - Iniially V cap (sc As h capacir chargs i vnually rachs h sam vlag as h bary r h EMF in his cas afr a vry lng im. This incras DOES NOT happn linarly. N: This is whil h capacir is CHARGING.

6 Currn Acrss h Capacir - Iniially I max =/R (sc Sinc h capacir is in SERIES wih h rsisr h currn will dcras as h pnial diffrnc bwn i and h bary apprachs zr. I is h pnial diffrnc which drivs h valu fr h currn. N: This is whil h capacir is CHARGING.

7 Tim Dmain Bhavir Th graphs w hav jus sn shw us ha his prcss dpnds n h im. L s lk hn a h UNITS f bh h rsisanc and capacianc. Uni fr Rsisanc = W = Vls/Amps Uni fr Capacianc = Farad = Culmbs/Vls R xc Vls Amps Culmbs Vls Culmb 1 Amp Sc Culmbs R xc Culmbs Scnds x Culmbs Amps SECONDS!

8 Th Tim Cnsan I is clar, ha fr a GIVEN valu f "C, fr any valu f R i ffcs h im ra a which h capacir chargs r dischargs. Thus h PRODUCT f R and C prduc wha is calld h CIUIT Capaciiv TIME CONSTANT. W us h Grk lr, Tau, fr his im cnsan. Th usin is: Wha xacly is h im cnsan?

9 Th Tim Cnsan Th im cnsan is h im ha i aks fr h capacir rach 63% f h EMF valu during charging.

10 Charging Bhavir Is hr a funcin ha will allw us calcula h vlag a any givn im? L s bgin by using KVL V cap (sc W nw hav a firs rdr diffrnial uain.

11 Charging funcin Hw d w slv his whn w hav 2 changing variabls? T g rid f h diffrnial w mus ingra. T mak i asir w mus g ur w changing variabls n diffrn sids f h uain and ingra ach sid rspcivly. R-arranging algbraically. Ging h cmmn dnminar Sparaing h numrar frm h dnminar, Crss muliplying. Sinc bh changing variabls ar n ppsi sid w can nw ingra.

12 Charging funcin 0 d C C ln( C C C C C ( 1 C C 0 d C (1 As i urns u w hav drivd a funcin ha dfins h CHARGE as a funcin f im. Hwvr if w divid ur funcin by a CONSTANT, in his cas C, w g ur vlag funcin. ( C C (1 C V ( (1

13 L s s ur funcin V V V ( (1 (1 (1 (1 ( Transin Sa Sady Sa V ( V V (3 ( Applying ach im cnsan prducs h charging curv w s. Fr pracical purpss h capacir is cnsidrd fully chargd afr 4-5 im cnsans( sady sa. Bfr ha im, i is in a ransin sa.

14 Charging Funcins ( V ( I( C (1 (1 I Charg and vlag build up a maximum whil currn fads zr Likwis, h vlag funcin can b dividd by anhr cnsan, in his cas, R, driv h currn charging funcin. Nw w hav 3 funcins ha allws us calcula h Charg, Vlag, r Currn a any givn im whil h capacir is charging.

15 Capacir Discharg Rsisr s Vlag Supps nw h swich mvs dwnwards wards h hr rminal. This prvns h riginal EMF surc b a par f h circui. V Rsisr A =0, h rsisr gs maximum vlag bu as h capacir cann kp is charg, h vlag drp dcrass. (sc

16 Capacir Discharg Rsisr s Currn Similar is charging graph, h currn hrugh h rsisr mus dcras as h vlag drp dcrass du h lss f charg n h capacir. I/R I Rsisr (sc

17 Capacir Discharg Capacir's Vlag Th discharging graph fr h capacir is h sam as ha f h rsisr. Thr WILL b a im dlay du h TIME CONSTANT f h circui. In his cas, h im cnsan is rachd whn h vlag f h capacir is 37% f h EMF.

18 Capacir Discharg Capacir s Currn Similar is charging graph, h currn hrugh h capacir mus dcras as h vlag drp dcrass du h lss f charg n h capacir. I/R I cap (sc

19 Discharging Funcins 0 1 IR V d cap d R d C d R d C d d 0 d Onc again w sar wih KVL, hwvr, h rasn w sar wih ZERO is bcaus h SOUE is nw gn frm h circui.

20 Discharging Funcins 1 d 1 0 d ln( Dividing ( by "C" hn "R" V ( I( I W nw can calcula h charg, currn, r vlag fr any im during h capacirs discharg.

21 Th bm lin ak away. Tim charg 63% = im cnsan au = τ = Whn charging a capacir ( V ( I( C (1 (1 I Charg and vlag build up a maximum whil currn fads zr. Whn discharging a capacir All hr fad away during discharg. V ( I( ( I I

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