Importnt Definitions nd Units Cpcitnce: CAPACITORS AND DIELECTRICS The property of system of electricl conductors nd insultors which enbles it to store electric chrge when potentil difference exists between the conductors. Units: frd (Coulomb per volt) Defined s the cpcitnce of cpcitor between the pltes of which there ppers potentil drop of 1 volt when it is chrged with one coulomb of electricity Dielectric: A non-conductor of electricity, insultor. A substnce in which n electric field gives rise to no net flow of electric chrge but only to displcement of chrge. Dielectric constnt or reltive permittivity: The rtio of the cpcitnce of cpcitor with the given substnce s the dielectric, to the cpcitnce of the sme cpcitor with ir (or vcuum) s the dielectric. Symbol κ. Dielectric strength: The mximum voltge which cn be pplied to dielectric mteril without cusing it to brek down. Usully expressed in volts per mm. Cpcitors Spheres The potentil of positively chrged conducting sphere is given by: V + ' = 1 4πε 0 R Where is the chrge on the sphere of rdius R. If nother sphere of the sme rdius R, crrying negtive chrge -, is locted t distnce >>R from the first sphere then it cn be sid tht both spheres re electriclly isolted. The potentil of the second sphere is therefore given by: V ' = 1 4πε 0 R The potentil difference between the two spheres is thus:
V' = V ' V ' = + 1 2 4πε 0 R The potentil difference is therefore proportionl to the chrge on either sphere. This eution cn now be written: = ( 2πε 0R) V' = CV ' ' C, the proportionlity constnt, is clled the cpcitnce of the two spheres. When the two spheres re moved closer together then our initil eutions, for the potentil of sphere, do not hold s they were derived with the ssumption tht the field round ech possessed sphericl symmetry - this cn no longer be the cse s the lines of force which emnted uniformly from the sphere, re now ffected by the presence of the second sphere. When positive chrge is brought close to n object then it hs the effect of rising tht objects potentil. When negtive chrge is brought close to n object then it hs the effect of lowering tht objects potentil. Thus, when the two spheres re brought closer together, the positive sphere will hve the effect of rising the potentil of the negtive sphere from V - to V - nd the negtive sphere will lower the potentil of the positive sphere from V + to V +. It is therefore simple to deduce tht, lthough the chrge on ech sphere hs remined constnt, the potentil difference between the spheres hs been reduced nd the cpcitnce hs been incresed. = CV where C>C nd V<V Prllel plte cpcitor This cpcitor is formed from two pltes with re A seprted by distnce d. When the pltes re connected to the terminls of bttery then positive chrge + will pper on one plte nd negtive chrge - will pper on the other. If we mke the ssumption tht d is smll compred to the re of the pltes then it cn be sid tht uniform field E exists between the pltes, with the lines of force being prllel nd evenly spced. -2-
Using Guss s lw, it is possible to clculte the cpcitnce of the device. The Gussin surfce is of height h with n re eul to tht of the pltes (shown on the digrm). The only prt of the Gussin surfce we re concerned with is tht which lies between the pltes s the surfce which lies within the plte hs zero field nd the two side surfces re t 90 to the field. For the surfce between the plte E is constnt nd therefore the flux is eul to EA Therefore: ε 0 φ = ε 0 EA = The work needed to tke test chrge 0 from one plte to the other is eul to 0 V or the force ( 0 E) multiplied by the distnce (d). These expressions must be eul i.e. 0 V = 0 Ed V = Ed Substituting into the stndrd cpcitor eution C=/V we get: C ε 0EA ε 0A = = = V Ed d Prllel plte cpcitor A cylindricl cpcitor A cylinder cpcitor is mde up of two coxil cylinders of rdius nd b nd of length l. We ssume tht the length l >> b nd construct Gussin surfce which is cylinder of rdius r nd length l, where b > r > Using Guss s lw: ε 0 = E.dA =Eε02 πrl E= 2πrlε 0-3-
The potentil difference between the pltes is: The cpcitnce is given by: b b dr V= Edr= = ln b 2πε 0l r 2πε 0l C= = 2πε l 0 V ln(b/ ) The cpcitnce only depends on the geometry of the cpcitor. If we look bck t the eution for prllel plte cpcitor it cn be seen tht the cpcitnce is lso only function of the dimensions of the device. Cpcitors in Series For cpcitors connected in series, the mgnitude of the chrge on ech of the pltes must be eul. This is due to the fct tht chrge is conserved nd the net chrge must therefore be zero for ech pir of pltes (whether they be from the sme cpcitor or different cpcitors). Applying the stndrd eution = CV to ech cpcitor: V 1 = /C 1 V 2 = /C 2 V 3 = /C 3 The totl potentil for the series combintion is: V = V 1 + V 2 + V 3 V = /C 1 + /C 2 + /C 3 = (1/C 1 + 1/C 2 + 1/C 3 ) The euivlent cpcitnce is therefore: C = /V = 1/(1/C 1 + 1/C 2 + 1/C 3 ) 1/C = 1/C 1 + 1/C 2 + 1/C 3 Cpcitors in Prllel -4-
When set of cpcitors re connected in prllel then it follows tht the voltge cross ech is constnt. The chrge on ech cpcitor will differ nd is given by = CV. The totl net chrge, before bttery is connected is zero, this is still the cse fter the bttery is connected but now the chrge hs seprted with ech cpcitor hving net chrge of zero. The totl chrge will be: = 1 + 2 + 3 = C 1 V + C 2 V + C 3 V = (C 1 + C 2 + C 3 ) V The euivlent cpcitnce C is: C = /V = (C 1 + C 2 + C 3 ) V/V C = C 1 + C 2 + C 3-5-