A capacitor is simply two pieces of metal near each other, separated by an insulator or air. A capacitor is used to store charge and energy.

Transcription

1 -1 apactors A capactor s smply two peces of metal near each other, separate by an nsulator or ar. A capactor s use to store charge an energy. A parallel-plate capactor conssts of two parallel plates separate by a stance, each plate wth area A. If A s large an s small, the plates are effectvely nfnte planes, an the -fel s unform an entrely n-between the plates. W L h V + on top plate area A L W lo V on top plate harges are always on the nse surfaces, because (+) attracts ( ). The outse surfaces reman uncharge. "harge on a capactor" always means + on one plate, on the other plate. apactors are charge by transferrng ( ) charge from one plate to the other. Takng ( ) charge off a plate leaves behn an equal-sze (+) charge. The charges make an -fel, whch means a voltage fference between the plates. The "voltage V on a capactor" always means the voltage fference V between the plates. V V V rato constant V It s always true that V, snce oubles.) An t s always true the, snce f V r (ouble the -fel everywhere an V q k r ˆ (ouble all the charges r everywhere an oubles). So the rato /V s always a constant: f you ouble the charge, the V ( V) s guarantee to ouble. Phys110 Dubson 9/18/009 Unversty of olorao at Bouler

2 - Defnton: capactance of a capactor: V If we ouble the charge, the voltage V oubles, but the rato /V remans constant. [Remember: means + an, V means V.] unts [] coulomb / volt fara (F) Bg capactance (1F) can store a bg wth a small V Small capactance (nf 10 9 F) small store wth a bg V For a parallel-plate capactor, wth ar or vacuum between the plates, the capactance s ε (ar or vacuum separatng plates) o A area A ε o ("epslon-naught") s the same constant that appeare n Gauss's Law. Proof: σ V V ε A ε Rearrangng, we get 0 0 ε. Done. V o A (We have use σ ε for a capactor.) Notce that the capactance of a parallel-plate capactor epens only on the sze an shape of the two metal parts. Ths turns out to true of all capactors. The capactance of two peces of metal epens solely on ther geometry. 0 Note that ths formula means ncreases as ecreases. Why? If s kept fxe, we have the same magntue -fel (because same charge ensty σ /A creates the σ/ε 0 ). Smaller an same-sze means smaller voltage V. Same an smaller V means bgger /V. smaller bgger Phys110 Dubson 9/18/009 Unversty of olorao at Bouler

3 -3 A fara s a huge capactance. For example, suppose we make a parallel plate capactor wth area A 1 m (bg) an separaton 1 mm m. The capactance s only 1 εo A ( )(1) F 9nF 3 10 (tny!) Mult-fara capactors n small packages are mae by makng very small. atomc mensons nm (nanometer) s possble. Store nergy n apactors It takes work to charge a capactor, because t s ffcult to transfer more electrons from the (+) plate to the ( ) plate. The work requre to transfer a charge q across a voltage fference "V" V s P q V. When we charge up a capactor from q ntal 0 to q fnal, we transfer electrons one at a tme. The frst electron s easy to transfer snce V V 0 ntally, but the later electrons take more an more work to transfer as (an V) buls up. q e (easy) e (har) Total work to charge capactor electrostatc potental energy store n capactor U 1 V (We use U for energy to avo confuson wth for electrc fel.) Why the (1/)? Why not P W ext V? Whle transferrng the total charge, the voltage fference ncrease from 0 to V. The average value was (1/)V. We can show ths more rgorously by ong an ntegral. When the voltage fference between the plates s V, the work requre to transfer an extra bt of charge q s U V q (q/) q. The total work ( total P) to charge the capactors s the sum (the ntegral) of the works one to transfer all the bts of charge: q U U q 0 an rewrte U n varous ways usng / V, V, V / : Phys110 Dubson 9/18/009 Unversty of olorao at Bouler

4 U V V Where s ths energy? The -fel contans the energy. It takes work to create an -fel. It turns out that the energy per volume (the energy ensty) of the -fel s gven by U 1 u o Vol. ε ε A "Proof": U V ( ) ε0 (A volume ) U 1 u 0 Vol. ε (Ths s a proof for the specal case of a parallel-plate capactor only, but the result turns out to be true always.) The energy U (1/)V of a charge capactor s n the -fel between the plates. If we pull the plates apart, keepng the charge fxe, we ncrease the volume whch contans -fel an the total energy ncreases. It was har to pull the plates apart, because opposte charges attract. The work we went nto creatng more -fel (same sze -fel over larger volume). It turns out that the work one to assemble a collecton of charges (W ext U q V) s equal to 1 the energy n the -fel create: U ( ε0 ) (volume ntegral) V apactors n parallel or n seres. Symbol for capactor : For capactors n parallel, Phys110 Dubson 9/18/009 Unversty of olorao at Bouler

5 -5 A bg s equvalent to two smaller se by se: "Proof" : For capactors n seres: seres seres apactors flle wth electrcs The capactance of a capactor can be ncrease by placng an nsulator ("electrc") between plates. The electrc s polarze by the charges on the plates. + polarze nsulator For fxe on the plates, the -fel between the plates s reuce when a electrc s nserte because the polarzaton charge on the electrc partally cancels the charge on the plates. smaller smaller V V, smaller V an same on plates larger / V Let's call the orgnal -fel 0 an the fnal, smaller, -fel. The orgnal -fel between the plates has been reuce by a mensonless factor calle K (K electrc constant): 0 K. The electrc constant K s greater than 1 an the value epens on the materal the nsulator s mae of. εo For a capactor flle wth a electrc: KA Phys110 Dubson 9/18/009 Unversty of olorao at Bouler