C 1 Capacitors 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 A parallel plate capacitor consists of two parallel plates separated by a distance d each plate with area A If A is large and d is small the plates are effectively infinite planes and the E field is uniform and entirely in between the plates W L Q on top plate hi V d d E area A L W lo V Q on top plate Charges are always on the inside surfaces because attracts The outside surfaces remain uncharged Charge Q on a capacitor always means Q on one plate Q on the other plate Capacitors are charged by transferring charge from one plate to the other Taking charge off a plate leaves behind an equal sized charge The charges make an E field which means a voltage difference between the plates The voltage V on a capacitor always means the voltage difference V between the plates D V E d V V E Q ratio Q constant V f v v E dr double the E field everywhere and V It is always true that V E since DV i v qi doubles And it is always true the E Q since E k 2 r i double all the charges ri i Phys1120 Dubson 1 14 2019 University of Colorado at Boulder C 2 everywhere and E doubles So the ratio Q V is always a constant if you double the charge Q the V V is guaranteed to double C Definition capacitance C of a capacitor Q V If we double the charge Q the voltage V doubles but the ratio Q V remains constant Remember Q means Q and Q V means V units C coulomb volt farad F Big capacitance 1F can store a big Q with a small V Small capacitance nF 10 9 F small Q stored with a big V area A For a parallel plate capacitor with air or vacuum d between the plates the capacitance is C eo A d air or vacuum separating plates o epsilon naught is the same constant that appeared in Gauss s Law d Proof DV V E Rearranging we get C s Q d d e0 A e0 e A Q o V d We have used E s for a capacitor e0 Done Notice that the capacitance of a parallel plate capacitor depends only on the size and shape of the two metal parts This turns out to true of all capacitors The capacitance of two pieces of metal depends solely on their geometry Note that this formula means C increases as d decreases Why If Q is kept fixed we have the same magnitude E field because same charge density Q A creates the E Smaller E smaller d E bigger C Phys1120 Dubson 1 14 2019 University of Colorado at Boulder C 3 d and same sized E means smaller voltage V E d Same Q and smaller V means bigger C Q V A farad is a huge capacitance For example suppose we make a parallel plate capacitor with area A 1 m2 big and separation d 1 mm 0 001 m The capacitance is only C eo A 8 85 10 12 1 d 10 3 9 10 9 F 9 nF tiny Multi farad capacitors in small packages are made by making d very small d atomic dimensions nm nanometer is possible Stored Energy in Capacitors It takes work to charge a capacitor because it is difficult to transfer more electrons from the plate to the plate The work required to transfer a charge q across a voltage difference V V is PE q V When we charge up a capacitor from qinitial 0 to qfinal Q we transfer electrons one at a time The first electron is easy to transfer since V V 0 initially but the later electrons take more and more work to transfer as Q and V builds up e hard q e easy Total work to charge capacitor electrostatic potential energy stored in capacitor U 1 QV 2 We use U for energy to avoid confusion with E for electric field Why the 1 2 Why not PE Wext Q V While transferring the total charge Q the voltage difference increased from 0 to V The average value was 1 2 V We can show this more rigorously by doing an integral When the voltage difference between the plates is V the work required to transfer an extra bit of charge dq is dU V dq q C dq Phys1120 Dubson 1 14 2019 University of Colorado at Boulder C 4 The total work total PE to charge the capacitors is the sum the integral of the works done to Q q Q2 dU dq transfer all the bits of charge U C 2C 0 Can rewrite U in various ways using C Q V Q C V V Q C U 1 1 1 Q2 Q V C V2 2 2 2 C Where is this energy The E field contains the energy It takes work to create an E field It turns out that the energy per volume the energy density of the E field is given by u Proof U U 1 eo E 2 Vol 2 e0 A 1 1 1 2 U 1 C V2 E d e0 E 2 A d u e0 E 2 2 2 d 2 Vol 2 volume This is a proof for the special case of a parallel plate capacitor only but the result turns out to be true always The energy U 1 2 QV of a charged capacitor is in the E field between the plates If we pull the plates apart keeping the charge Q fixed we increase the volume which contains E field and the total energy increases It was hard to pull the plates apart because opposite charges attract E E The work we did went into creating more E field same size E field over larger volume It turns out that the work done to assemble a collection of charges Wext U q V is equal to 12 e0 E 2 dV the energy in the E field created U volume integral Capacitors in parallel or in series Symbol for capacitor Phys1120 Dubson 1 14 2019 University of Colorado at Boulder C 5 For capacitors in parallel C C1 C2 C1 C2 C A big C is equivalent to two smaller C side by side Proof For capacitors in series Cseries 1 1 1 C1 C2 C1 C2 Cseries Capacitors filled with dielectrics The capacitance C of a capacitor can be increased by placing an insulator dielectric between plates The dielectric is polarized by the charges on the plates Q polarized insulator Q For fixed Q on the plates the E field between the plates is reduced when a dielectric is inserted because the polarization charge on the dielectric partially cancels the charge Q on the plates smaller E smaller V V E d smaller V …
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