EMF 2005 Handout 5: Capacitance 1 CAPACITANCE. Q Coulombs. Volts
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1 MF 005 Hanut 5: apacitance APAITAN Definitin f capacitance Recall: Fr a pint charge r a charge sphere V 4πε r In general, POTNTIAL HARG fr any size r shape f cnuctr. Definitin: The cnstant f prprtinality between V an is calle APAITAN, : V Units f capacitance: ulmbs Vlts V Definitin: Fara (F) V apacitance is a measure f the ability f a cnuctr r a system f cnuctrs t stre charge (an hence t stre energy). Recall: Fr a charge cnuctr U tt V U tt V r U tt Relatinship between capacitance an energy Alternative units fr ε : ulmbs 4πε r ε metres Vlts metres ε ulmbs (Vlts)(metres) Faras metres ε Fm These are the units in which ε is usually qute.
2 MF 005 Hanut 5: apacitance Prceure fr fining capacitance. Imagine a charge n the cnuctr (if it s a single cnuctr) r charges f ± (fr a pair f cnuctrs). Fin (e.g., use Gauss s Law) 3. Fin the ptential (ifference) using V L (never min the sign). 4. Put /V [ always cancels ut] a b Nte:. epens nly n the gemetry the size an shape f the cnuctrs an the istance between them.. is inepenent f [because V ] xamples:. apacitance f an islate sphere. Parallel plate capacitr 3. apacitance f tw cncentric spheres 4. apacitance per unit length f a caxial cable See lecture ntes
3 MF 005 Hanut 5: apacitance 3 apacitr An electrical cmpnent esigne t have a particular value f. apacitrs in parallel is the same fr bth: tt tt tt tt tt tt Therefre tt tt V apacitrs in series tt tt In this case is the same fr bth an is ifferent. V V tt tt tt tt tt
4 MF 005 Hanut 5: apacitance 4 Dielectrics an plarisatin Until nw, we have assume that the space between charges, cnuctrs, etc. is empty (a vacuum). What if it s fille with sme insulating material? Recall: A DILTRI (insulatr) is electrically neutral. But it cntains many ve an ve charges in its atms r mlecules. Because it s an insulatr, the charges can t mve arun. BUT: sme mlecules have a natural DIPOL MOMNT when place in an external electric fiel, the iples ten t align with it. iple Trque ven if n INTRINSI iples exist, they can be INDUD by an applie fiel, prucing the same effect. nsier a parallel plate capacitr, with psitive an negative charges n the plates. This creates an electric fiel between the plates. If there is a ielectric between the plates, then ue t iple alignment, r (POLARISATION) the istributins f ve an ve charge nt verlap exactly: xcess ve charge n the right. xcess ve charge n the left. The inuce (POLARISD) fiel, i, tens t OPPOS. The resultant fiel is LSS than the applie fiel: tt i i
5 MF 005 Hanut 5: apacitance 5 Dielectric cnstant tt i Let tt (K ) K K (smetimes written as the greek letter κ kappa) is the DILTRI ONSTANT r RLATIV PRMITTIVITY f the material. Let σ charge ensity n capacitr plates: Let σ i inuce charge ensity n surfaces f ielectric: σ /ε i σ i /ε Therefre σ ε ε [ ] i i tt ε K S the inuce surface charge ensity is σ i σ K The effect f ielectric cnstant n capacitance nsier the case f a parallel plate capacitr. N ielectric: εa With ielectric: K KεA apacitance with ielectric K( apacitance withut ielectric) S, capacitance (i.e., the ability t stre charge an hence energy) is increase by the use f a ielectric.
6 MF 005 Hanut 5: apacitance 6 Typical values f ielectic cnstant K Air (nt much ifferent frm a vacuum) Plythene.3 Glass 5 0 Germanium 6 Water 80 A perfect cnuctr? Gauss's Law in ielectrics Just replace ε with Kε : Φ A Kε enclse Dielectric breakwn If > sme limit, the BRAKDOWN FILD STRNGTH f the material, then the bns between the electrns an the atms are brken an the material becmes a ONDUTOR ( shrt circuit in a cable r capacitr).
7 MF 005 Hanut 5: apacitance 7 apacitance per unit length f a pair f caxial cyliners (e.g., a caxial cable) a b Step : Let charge per unit length be λ n inner cnuctr λ n uter cnuctr Step : Fin : Fiel pattern: Fiel lines g frm ve charges n inner cnuctr t ve charges n uter cnuctr 0 fr r < a an r > b (A Gaussian cyliner enclses n charge) λ Apply Gauss s Law πεr (similar t a previus example) Step 3: Fin : L Path: Integrate alng a fiel line L an L is parallel t s L r b λ λ b r ln πε r πε a a r r b L r a r 0 We take t be psitive when fining capacitance. Step 4: Put /V πε (apacitance/unit length) b ln a
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