Energy stored in capacitor

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Lizbeth Harmon
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1 Energy sored in capacior V = V B harging capacior + Q V = V B - Q V = V B Sored charge + Q V = V B - Q V = V B Sored charge produces he curren + Q - Q V = V B 10mA R

2 Energy sored in capacior Elecric Energy = harge x Volage: W = Q V This formula would be rue for capaciors, had he charge beween plaes been ransferred a a consan volage. For capaciors, Q = V As he charge Q on he plae increases, he volage V increases oo (and vice versa). Suppose he capacior was charged o he volage V 0 : Q Q 0 W V 0 The sored energy is he area under he Q V line. V V Q W = Noe ha, Q 0 = V 0 : 2 W = V V 0 is he volage on he charged capacior 2 0

3 Example problem 1 The capacior of 1 mf has been charged o 100 V. Wha energy is sored in he capacior in Joules? V W = 0 of Timed response

4 ommercial capaciors ε = d ε d 0 A 20 mf capacior

5 apacior bank (series-parallel)

6 Parallel connecion of capaciors V = V 1 2 B Q 1 = 1 V 1 Q 2 = 2 V 2 apaciors 1 and 2 are conneced in parallel: boh erminal of each capacior are conneced o he same wires. The volage on each of he capaciors is he same, V B The charge on he capacior 1, Q 1 = 1 V B The charge on he capacior 2, Q 2 = 2 V B Toal charge sored in boh capaciors: Q To = Q 1 + Q 2 = ( ) V B The equivalen capaciance, eq = Q To / V B = par = 1 + 2

7 V = V B V = V B 1 2 Parallel connecion of capaciors We can also find he equivalen capaciance from he KL. For he 1 s and 2 nd capaciors, ; ; V I V I = = I T I 1 I 2 According o he KL, he oal curren I T = I 1 + I 2. Subsiuing he values for I 1 and I 2 : ( ) ; V V V V I par T = + = + = par = 1 + 2

8 Series connecion of capaciors V = V 1 B 2 Q = V ; Q = V ; V 1 V 2 apaciors 1 and 2 are conneced in series: he charge Q on 1 and 2 is he same; For each of he capaciors, From he KVL, V To = V 1 + V 2. V = V The oal charge, on he 1 2 combinaion is sill Q. The equivalen capaciance is defined as: Q = EqS V To 1 Q = EqS ( V1 + V2 ) = EqS V1 + V1 ; Q = 1 V1 ; 2 1/ Ser = 1/ 1 + 1/ = S EqS = 1; 2

9 Example problem 2 Three capaciors 2 mf each are conneced in parallel. Wha is he oal capaciance in mf? 0 of Timed response

10 Example problem 3 Three capaciors 2 mf each are conneced in series. Wha is he oal capaciance in mf? 0 of Timed response

11 Example problem 4 Parallel-plae capacior has a capaciance of 9 nf. A hin meal plae has been insered in he middle beween he op and boom plaes. Wha is he capaciance (in nf) of he capacior now? εdε0 A = d 0 of Timed response

12 Transiens in R- circui V B Series R- circui R I = I R = d V d V R R The firs momen afer closing he swich, he volage across he capacior = 0; The capacior behaves as a shor-circui; The curren a =0, I 0 = V B /R; Afer all he ransiens are over ( Yh), I = 0

13 ommuaion rule for capaciors V B R onsider a capacior righ before and righ afer commuaion in an arbirary circui. The capacior volage (charge) does no have o be zero before he commuaion. V_ V V + ommuaion even ime If V changes insananeously afer he commuaion, he curren in he conneced circui would be infiniely high: d V V V I = = d d + 0 If V + is differen from V - when d 0, hen I The capacior volage does no change afer commuaion: V - = V +

14 Graphs showing he curren and volage for a capacior charging I( ) = V B R R e apacior volage V ( ) = VB 1 e τ R = R R When = 3 τ R, V = 0.95V B ;

15 Graphs showing he curren and volage for a capacior discharging apacior saring volage is V B I( ) = V B R R e V ( ) = V e B R τ R = R When = 3 τ R, V = 0.05V B =5% V B

16 General formula for sep response of an arbirary R- circui V S R R- circui ( ) / v = v + v v e τ τ = R F 0 F V 0 is he capacior volage righ afer (or righ before) he commuaion; V F is he capacior volage long ime afer all he ransien processes are over. R is he oal resisance conneced o he capacior afer commuaion (al he sources are zeroed o find he equivalen oal resisance)

17 V B =4.5 V Example 1 Delayed alarm circui S1 R=10 k =0.5 mf S2 Sensor swich: S1. Elecronic swich S2 riggers he alarm sysem when he volage across i exceeds he prese hreshold value V T. R = 10k; = 0.5 mf; V B = 4.5 V. Assume he S2 resisance infiniely high. The required ime delay beween he swich S1 urn-on and riggering swich S2 mus be = 3s. Wha hreshold volage V T mus he swich S2 be uned o?

18 V B =4.5 V Example 1 Delayed alarm circui S1 R=10 k =0.5 mf S2 v = v + v v e τ ( ) / F 0 F V 0 = 0; V F = V B ; τ = R ; / v vb vb e τ = The required alarm riggering ime: =3s The required swich hreshold volage v SW = v c ( r ): 3/( ) e e vsw = e = e = 2V

19 Example 2 The swich in he circui shown in Fig has been in posiion a for a long ime. A = 0 he swich is moved o posiion b. Wha is he v ime dependence a >0? v = v + v v e τ ( ) / F 0 F v 0 = V 60Ω = 40V *60/(60+20) = -30 V v F = 90 V; τ = R = 400 kω 0.5µF = 0.2 s. / 0.2 v = e V

RC (Resistor-Capacitor) Circuits. AP Physics C

RC (Resistor-Capacitor) Circuits. AP Physics C (Resisor-Capacior Circuis AP Physics C Circui Iniial Condiions An circui is one where you have a capacior and resisor in he same circui. Suppose we have he following circui: Iniially, he capacior is UNCHARGED