How to get mechanical work from a capacitor and a couple of batteries
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1 How to get mechancal work rom a capactor and a couple o batteres E. N. Mranda Área de Cencas Eactas CRICYT 55 Mendoza, rgentna Facultad de Ingenería Unversdad de Mendoza 55-Mendoza, rgentna Departamento de Físca Unversdad Naconal de San Lus 57-San Lus, rgentna bstract: The work done by a parallel plate capactor s evaluated when the plate separaton s changed. Two cases are consdered: 1) the capactor has a constant charge; ) the capactor s at constant voltage. The net work s calculated when the devce ollows a closed cycle n the charge-voltage space. For certan condtons a net mechancal work can be obtaned rom the cyclng capactor. The analyss s smple enough to be eplaned n a general physcs course. PCS: 41., 84.3, 84.5
2 I we thnk about gettng mechancal work rom a battery, an electrcal engne comes mmedately to mnd. Ths s a natural assocaton snce t s the usual way o transormng electrochemcal energy stored n the battery nto mechancal work [1, ]. However, n ths artcle an alternatve way wll be shown. It s a novel approach usng a capactor and a couple o batteres wth derent voltages. The analyss o the system should be understandable by reshman students wth a good background n general physcs [3, 4]. We wll consder n our analyss a parallel plate capactor to get eact results; however, the general dea should work or any shape. The area o the capactor plates s, the gap between them, lled wth ar, s so the electrc permeablty o the delectrc medum s. Our rst step s to study the work done by a capactor wth a ed stored charge and a varable gap between ts plates. The physcal stuaton s the ollowng. capactor s ntally uncharged and the plate gap s. battery wth a voltage s connected to the capactor, and t gets a charge q. terwards, t s dsconnected rom the battery but remans charged; the plate space s changed to, and work has to be done on the capactor n order to mody the gap; our am s to evaluate that work W q. The subnde q means the system s wth constant charge. To calculate W, t s sucent to evaluate the change n the capactor potental energy E wrtten as E q/c where C s the capactance. For a parallel plate capactor, the capactance s C / and we can wrte: W q E 1 q C E 1 1 q ( q C ) (1) Eq. (1) eplctly states that the stored charge remans constant n the process. However, t would be useul to restate t n terms o the battery voltage. One should remember that C q/; consequently, or a plane capactor one gets: Wq ( ) () We have ound the work done by a parallel plane capactor that has been connected to a battery wth voltage, got a charge q, dsconnected rom the battery and ts plate separaton changed rom to. Snce the process s at constant charge q C C and:
3 Ths relaton wll be useul later. (3) The net step s to evaluate the work done when the voltage s constant. The capactor s connected to a battery wth voltage, and the plate separaton s changed rom to. One should remember that the orce among the plates can be wrtten as: E F 1 (4) The work needed to change the plate gap comes to be: W 1 1 The subnde means the process has taken place at constant voltage. Fd (5) We have all the elements requred to use the capactor as a devce that can delver mechancal work. The capactor has to ollow the cycle showed n Fgure 1. The analogy wth thermodynamc cycles o thermal engnes leads us to use a smlar termnology o epansons and compressons. The capactor starts at the state wth coordnates ( 1, 1 ) and s epanded at constant charge to B(, ); then, t s agan epanded at constant voltage rom B(, ) to C( 3, ). From C, t s compressed at constant charge untl t reaches the orgnal voltage n D( 4, 1 ); nally, the cycle s closed wth a compresson rom D( 4, 1 ) to. Our task s to evaluate each step to get the net mechancal work done (or receved) by the capactor. From to B, the work s at constant charge; then, accordng to eq. (1): 1 W, B ( 1 ) (6) 1
4 From B to C, the system s at constant potental; ollowng eq. (5), the work s: W B, C (7) From C to D, one gets: Fnally, rom D to, t s W C, D W D, 1 ( 4 3 ) (8) (9) 4 1 The net work s the sum o equatons (6)-(9). It should be notced that and 4 can be wrtten, accordng to (3), as: 1 / 1, /. The nal outcome s: W net (1) 1 ( ) Eq. (1) s the central result o ths paper and shows that the net work done by the capactor perormng the cycle shown n Fgure 1 s derent rom zero. One may get mechancal work / 3 > 1 / 1. graphcal nterpretaton o the prevous result s possble. The cycle has to be plotted n the (q, ) space as shown n Fgure. In ths space the process plot s a rectangle and the enclosed area s easy to evaluate. One can wrte: q1 C (11) q1 C3 3 Consequently, the rectangle area s: rea ( ) (1) 1 The area enclosed by the rectangle s the net work done by the system wth the opposte sgn. Our am s accomplshed: we can get net work rom a capactor ollowng the cycle o Fgure 1 or and shown by eq. (1) or (1) the relaton between 1,, 1 and 3 s the correct one.
5 It s pontless to evaluate the ecency o the system analyzed here snce all the work delvered by the capactor comes rom the batteres wth voltages 1 and. It would be more realstc to consder the wre resstance and get the Joule heat dsspated. However, t s dcult to evaluate the current lowng between the batteres and the capactor because ts capactance changes wth the plate separaton. One should know the temporal evoluton o the gap (t) to get the tme dependence o the capactance C(t). To summarze, an engne mght be bult rom two batteres wth potentals 1 and and a capactor. lthough o lttle technologcal mport, t s nterestng rom a conceptual pont o vew and aords a useul eercse or a general physcs course. cknowledgment: The author thanks partal nancal support rom the Natonal Scentc and Technologcal Research Councl o rgentna (CONICET).
6 Reerences: [1] Retz J. R., Mlord F.J., Chrsty R. W.; Introducton to Electromagnetc Theory; ddson Wesley, 199. [] Purcell. E.; Electrcty and Magnetsm; Berkeley Physcs Course vol. II; McGraw-Hll, 197. [3] Hallday D., Resnck R., Walker J.; Fundamentals o Physcs; Wley, 4. [4] Serway R.., Faughn J. S., ulle, C., Benett, C..; College Physcs; Brooks Cole, 5.
7 Fgure captons: Fgure 1: Ths graph shows the closed cycle ollowed by the system. The capactor s connected to a battery wth potental 1 at pont and the plate gap s 1. terwards, the battery s dsconnected and the plate separaton s ncreased, keepng constant the charge n the capactor. Once the voltage among the plates s, the capactor s connected to a battery wth that potental. Wth the capactor at a constant voltage, the plate separaton s ncreased agan up to 3, reachng pont C. Then, the plate gap s decreased untl the orgnal voltage 1 s reached at pont D. From ths pont, a new compresson at constant voltage takes place and the orgnal plate separaton s reganed. Notce that the slope o B (CD) s proportonal to the charge stored n the capactor. Fgure : The same cycle depcted n the prevous gure s shown n the (q, ) plane. Snce each cycle step s at constant charge or at constant voltage, the cycle shape s rectangular. It s a smple eercse to evaluate the area o the rectangle that comes to be mnus the net work perormed by the system.
8 v B C v 1 D Fgure 1 X v B C v 1 D q 1 q 3 q Fgure
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