Chapter 16. Capacitance. Capacitance, cont. Parallel-Plate Capacitor, Example 1/20/2011. Electric Energy and Capacitance

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1 summary C = ε A / d = πε L / ln( b / a ) ab C = 4πε 4πε a b a b >> a Chapter 16 Electric Energy and Capacitance Capacitance Q=CV Parallel plates, caxial cables, Earth Series and parallel = C = C1 + C +... cmbinatins C C1 C 1 Q Energy in a capacitr U = CV = C Dielectrics C κc Dielectric strength Capacitance A capacitr is a device used in a variety f electric circuits The capacitance, C, f a capacitr is defined as the rati f the magnitude f the charge n either cnductr (plate) t the magnitude f the ptential difference between the cnductrs (plates) Capacitance, cnt C Q V Units: Farad (F) 1 F = 1 C / V A Farad is very large C ε L Often will see µf r pf ε = 8.85x10-1 F/m V is the ptential difference acrss a circuit element r device V represents the actual ptential due t a given charge at a given lcatin Parallel-Plate Capacitr The capacitance f a device depends n the gemetric arrangement f the cnductrs Fr a parallel-plate capacitr whse plates are separated by air: A C = ε d Parallel-Plate Capacitr, Example The capacitr cnsists f tw parallel plates Each have area A They are separated by a distance d The plates carry equal and ppsite charges When cnnected t the battery, charge is pulled ff ne plate and transferred t the ther plate The transfer stps when V cap = V battery 1

2 Electric Field in a Parallel- Plate Capacitr The electric field between the plates is unifrm Near the center Nnunifrm near the edges The field may be taken as cnstant thrughut the regin between the plates Applicatins f Capacitrs Camera Flash The flash attachment n a camera uses a capacitr A battery is used t charge the capacitr The energy stred in the capacitr is released when the buttn is pushed t take a picture The charge is delivered very quickly, illuminating the subject when mre light is needed Applicatins f Capacitrs Cmputers Cmputers use capacitrs in many ways Sme keybards use capacitrs at the bases f the keys When the key is pressed, the capacitr spacing decreases and the capacitance increases The key is recgnized by the change in capacitance Capacitrs in Circuits A circuit is a cllectin f bjects usually cntaining a surce f electrical energy (such as a battery) cnnected t elements that cnvert electrical energy t ther frms A circuit diagram can be used t shw the path f the real circuit Capacitrs in Parallel When cnnected in parallel, bth have the same ptential difference, V, acrss them Capacitrs in Parallel When capacitrs are first cnnected in the circuit, electrns are transferred frm the left plates thrugh the battery t the right plate, leaving the left plate psitively charged and the right plate negatively charged The flw f charges ceases when the vltage acrss the capacitrs equals that f the battery The capacitrs reach their maximum charge when the flw f charge ceases

3 Capacitrs in Parallel The ttal charge is equal t the sum f the charges n the capacitrs Q ttal = Q 1 + Q The ptential difference acrss the capacitrs is the same And each is equal t the vltage f the battery Mre Abut Capacitrs in Parallel The capacitrs can be replaced with ne capacitr with a capacitance f C eq The equivalent capacitr must have exactly the same external effect n the circuit as the riginal capacitrs Capacitrs in Parallel, final C eq = C 1 + C + The equivalent capacitance f a parallel cmbinatin f capacitrs is greater than any f the individual capacitrs Capacitrs in Series When in series, the capacitrs are cnnected endt-end The magnitude f the charge must be the same n all the plates Capacitrs in Series When a battery is cnnected t the circuit, electrns are transferred frm the left plate f C 1 t the right plate f C thrugh the battery As this negative charge accumulates n the right plate f C, an equivalent amunt f negative charge is remved frm the left plate f C, leaving it with an excess psitive charge All f the right plates gain charges f Q and all the left plates have charges f +Q Mre Abut Capacitrs in Series An equivalent capacitr can be fund that perfrms the same functin as the series cmbinatin The ptential differences add up t the battery vltage 3

4 Capacitrs in Series, final V = V1 + V = + +K C C C eq 1 The equivalent capacitance f a series cmbinatin is always less than any individual capacitr in the cmbinatin Prblem-Slving Strategy Be careful with the chice f units Cmbine capacitrs fllwing the frmulas When tw r mre unequal capacitrs are cnnected in series, they carry the same charge, but the ptential differences acrss them are nt the same The capacitances add as reciprcals and the equivalent capacitance is always less than the smallest individual capacitr Prblem-Slving Strategy, cnt Cmbining capacitrs When tw r mre capacitrs are cnnected in parallel, the ptential differences acrss them are the same The charge n each capacitr is prprtinal t its capacitance The capacitrs add directly t give the equivalent capacitance Redraw the circuit after every cmbinatin Prblem-Slving Strategy, final Repeat the prcess until there is nly ne single equivalent capacitr A cmplicated circuit can ften be reduced t ne equivalent capacitr Replace capacitrs in series r parallel with their equivalent Redraw the circuit and cntinue T find the charge n, r the ptential difference acrss, ne f the capacitrs, start with yur final equivalent capacitr and wrk back thrugh the circuit reductins Prblem-Slving Strategy, Equatin Summary Circuit Reductin Example Use the fllwing equatins when wrking thrugh the circuit diagrams: Capacitance equatin: C = Q / V Capacitrs in parallel: C eq = C 1 + C + Capacitrs in parallel all have the same vltage differences as des the equivalent capacitance Capacitrs in series: 1/C eq = 1/C 1 + 1/C + Capacitrs in series all have the same charge, Q, as des their equivalent capacitance 4

5 Energy Stred in a Capacitr Energy stred = ½ Q V Frm the definitin f capacitance, this can be rewritten in different frms 1 1 Q Energy = Q V = C V = C Applicatin Defibrillatrs When fibrillatin ccurs, the heart prduces a rapid, irregular pattern f beats A fast discharge f electrical energy thrugh the heart can return the rgan t its nrmal beat pattern In general, capacitrs act as energy reservirs that can be slwly charged and then discharged quickly t prvide large amunts f energy in a shrt pulse Capacitrs with Dielectrics Capacitrs with Dielectrics A dielectric is an insulating material that, when placed between the plates f a capacitr, increases the capacitance Dielectrics include rubber, plastic, r waxed paper C = κc = κε (A/d) The capacitance is multiplied by the factr κ when the dielectric cmpletely fills the regin between the plates Dielectric Strength Fr any given plate separatin, there is a maximum electric field that can be prduced in the dielectric befre it breaks dwn and begins t cnduct This maximum electric field is called the dielectric strength 5

6 An Atmic Descriptin f Dielectrics Plarizatin ccurs when there is a separatin between the centers f gravity f its negative charge and its psitive charge In a capacitr, the dielectric becmes plarized because it is in an electric field that exists between the plates Mre Atmic Descriptin The presence f the psitive charge n the dielectric effectively reduces sme f the negative charge n the metal This allws mre negative charge n the plates fr a given applied vltage The capacitance increases summary C = ε A / d = πε L / ln( b / a ) ab C = 4πε 4πε a b a b >> a Applicatin Electrstatic Precipitatr Capacitance Q=CV Parallel plates, caxial cables, Earth Series and parallel = C = C1 + C +... cmbinatins C C1 C 1 Q Energy in a capacitr U = CV = C Dielectrics C κc Dielectric strength It is used t remve particulate matter frm cmbustin gases Reduces air pllutin Can eliminate apprximately 90% by mass f the ash and dust frm smke Recvers metal xides frm the stack Applicatin Electrstatic Air Cleaner Used in hmes t reduce the discmfrt f allergy sufferers It uses many f the same principles as the electrstatic precipitatr Applicatin Xergraphic Cpiers The prcess f xergraphy is used fr making phtcpies Uses phtcnductive materials A phtcnductive material is a pr cnductr f electricity in the dark but becmes a gd electric cnductr when expsed t light 6

7 The Xergraphic Prcess Applicatin Laser Printer The steps fr prducing a dcument n a laser printer is similar t the steps in the xergraphic prcess Steps a, c, and d are the same The majr difference is the way the image frms n the selenium-cated drum A rtating mirrr inside the printer causes the beam f the laser t sweep acrss the seleniumcated drum The electrical signals frm the desired letter in psitive charges n the selenium-cated drum Tner is applied and the prcess cntinues as in the xergraphic prcess 7