UNIVESITY OF TECHNOLOGY, SYDNEY FACULTY OF ENGINEEING 4853 Electmechanical Systems Electstatics Tpics t cve:. Culmb's Law 5. Mateial Ppeties. Electic Field Stength 6. Gauss' Theem 3. Electic Ptential 7. Capacitance 4. Flux & Flux Density 8. Electstatic Enegy Intductin The cncept f a field is used t descibe "actin at a distance" a distubance input at ne pint can have an effect utput at a distant pint. The egin whee the effect f this cupling media is felt is the field, descibed by its (vect) field stength. Tw kind f electic chages (,) ae knwn t exist. The electstatics is a they f the inteactin f statinay electic chages in a medium. Culmb's Law (785) The fce expeienced by a static chage due t the adial field f can be witten as: F whee ε is the pemittivity f the medium and is a unit vect in the diectin fm t. F fee space, εε whee ε 0 9 /36π. F Electic Field The egin in which fces ae expeienced due t the pesence f electic chages is called an electic field. At all pints within this egin the electic field stength will have a magnitude and a diectin. E, the electic field stength, is defined at any pint t be the fce acting n a unit psitive chage placed at that pint (with the pvis that the intductin f the unit chage in n way distubs the distibutin f the existing chages in the field). Theefe, the electstatic fce acting n a chage can be expessed as F E
Electic Field Stength Due t a single pint chage E ( Vm ) Due t a system f discete chages Due t a cntinuus vlume chage distibutin Due t a cntinuus suface chage distibutin Due t a cntinuus line chage distibutin E n k k E E E k V' k ρ ρ s S' ρ l L' dv' ds' dl' Example - Field due t Distibuted Chages Detemine the electic field stength E at a pint P a distance d abve an infinite chaged plane. The density f chage n the plane is culmbs pe squae mete. P d dθ d ds S Slutin - Field due t Distibuted Chages The fce n a unit psitive chage placed at pint P due t the chage n ds is dθd/ (d ) in the diectin f SP. This can be eslved int tw thgnal cmpnents: dθ d d 0 ( d ) d at ight angles t the plane and dθ d 0 ( d ) d paallel t the plane If a full ing f chage is cnsideed then it is clea that the paallel cmpnents will cancel and the fce will nly have a nmal cmpnent. Slutin - Cnt. The fce nmal t the plane due t an incemental ing f chage is π d 0 d 3 d d ε ( ) 0 d ( d ) The ttal fce is fund by summing the cntibutins f all incemental ings. Theefe, the magnitude f E is E d d d 3 ε0 ε ( d ) d 0 0 ε 0 0 E n ε 0 3
Electic Ptential Diffeence The electic ptential diffeence between tw pints in a field is defined as the extenal wk (against the field stength vect) needed t mve a unit psitive chage fm ne pint t anthe. The wk dne by the field in distance dl is dw qe dl and q C Theefe, the ptential diffeence between tw pints V ba b b a E d l a dw Electic Ptential The ptential diffeence is a scala, and it is independent f the integatin path. Futheme, if we make sme abitay chice f pint (pssibly, but nt necessaily, at ) at which the ptential is taken t be ze then we can attach a unique value f ptential t evey the pint in the field. The unit f ptential diffeence is Vlts (V). It is als nted that dv E dl ( Vm ) Electic Flux Lines Electic flux lines ae used t visulize static electic fields. Electic Flux and Flux Density The electic flux density, D, (Cm ) is the amunt f flux pe unit nmal aea: δψ dψ D lim δa 0 δa da whee ψ is the electic flux, and ψ D da In fee space, the electic flux density D is elated t the electic field stength E as D ε E
Cnducts in Static Electic Field Inside a cnduct, electic chages ae fee t mve. Cnduct Cnduct (a) Cnduct E 0 ρ 0 (c) Cnducts in Static Electic Field - Cnt. Unde static cnditins, inside a cnduct, the vlume chage density ρ 0, and E 0. E the cnduct suface, i.e. the suface f a cnduct is an equiptential suface. (b) Dielectics in Static Electic Field - Induced Electic Diples Ideal dielectics d nt cntain fee chages. The pesence f an extenal electic field causes a fce t be exeted n each chaged paticle and esults in small displacements f and chages in ppsite diectins. These displacements plaize a dielectic and ceate electic diples. Dielectics in Static Electic Field - Electets The mlecules f sme dielectics pssess pemanent diple mments, even in the absence f an extenal plaizing field. Such mateials ae called electets. Electets ae the electical equivalents f pemanent magnets; they have fund imptant applicatins in high fidelity electet micphnes.
Dielectics in Static Electic Field - Electic Hysteesis If E vaies peidically, the vaiatin f D lags that f E. This is knwn as electic hysteesis f an dielectic. The electic hysteesis lss (aea f the D-E lp) can be calculated by P hyst E d D Dielectics in Static Electic Field - Dielectic Cnstants When the electic hysteesis f a dielectic is igned and the dielectic ppeties ae egaded as istpic and linea, the plaizatin is diectly pptinal t the electic field stength, and the pptinality cnstant is independent f the diectin f the field. We wite D εe whee εε ε is the abslute pemittivity, simply pemittivity, and ε a dimensinless quantity knwn as the elative pemittivity the dielectic cnstant. Gauss Theem Gauss' theem states that f any clsed suface the ttal utwad flux is equal t the algebaic sum f all the fee chages enclsed within the suface. Mathematically stated that is ψ D da q A enclsed Example - A Spheical Cnduct in Fee Space Cnside a spheical cnduct in fee space, adius, with unifm suface chage q Cm -. Detemine the electic field stength at a pint at distance fm the cente f the sphee. Cnside tw cases: > and <. (Assume that thee ae n the chaged bdies in the vicinity, which is implied by the Gaussian suface unifm chage distibutin.) Slutin: Cnstuct an imaginay sphee f adius, cncentic with the cnduct (Gaussian suface) and apply Gauss' theem t this (clsed) suface. By symmety, E and D will be cnstant ve this suface and adially diected. 0 cnducting sphee
Example - A Spheical Cnduct in Fee Space Slutin (cnt.): F the case >, the chage enclsed by the Gaussian suface is: suface aea f the cnducting sphee suface chage pe unit aea, that is q 4π q enclsed The suface integatin f D is D da D da 4π D A Theefe, q enclsed q E n n 4π ε ε whee n is the unit utwad adial vect. F the case <, when the enclsed chage is ze and theefe E 0 Execise A spheical cnduct with unifm suface chage q Cm - and cated in a dielectic mateial with ε ε is placed in fee space. Assume the adius f the cnduct sphee is and the thickness f the dielectic cating is d. Detemine the field stength at a pint inside the cating and at a pint utside the cated sphee. Example - An Infinitely lng unifmly chaged cnduct in fee space In this case the chage is unifmly distibuted alng the suface f the cnduct with Cm -. Detemine the electic field stength at a pint at distance fm the axis f the cnduct. Slutin: Gaussian suface The symmety (implied by the infinite length and staightness f the cnduct) ensues that the 0 field stength at a pint distant fm the cnduct axis will be adial. As a Gaussian suface we chse a cncentic unit length cylinde. The chage enclsed is theefe. infinitely lng cnducting cylinde Example - An Infinitely lng unifmly chaged cnduct in fee space Slutin (cnt.): The adial natue f E (and theefe D) means that n flux passes thugh the plane cicula sides f the Gaussian suface. D is cnstant n the cuved pat f the Gaussian suface. Als the angle between the vects D and da is 0. Theefe Gauss' theem in this case leads t D da D da πd A Theefe, E n πε0 whee, as befe, n is the unit utwad adial vect.
Execise Detemine the electic field at a pint P a distance d abve an infinite chaged plane using Gauss' theem. Assume the density f chage n the plane is Cm -. Cmpae yu answe with the answe btained ealie using Culmb's Law. Capacitance - Definitin Cnside an islated cnduct f any shape and place chage n it then its electic ptential will ise. Assume that it ises t V vlts. Nw we place anthe chage f Culmb n it. Using supepsitin the cnduct's ptential ises t V vlts. That is, V. Define the pptinality cnstant, C, as the capacitance f the islated bdy, CV The unit f capacitance is CV - F. Capacitance - Calculatin The diagam belw shws a capacit, which cnsists f tw cnducts f abitay shapes sepaated by fee space a dielectic medium. When a dc vltage is applied t the cnducts, a chage tansfe ccus, esulting in a chage f n ne cnduct and n the the. Capacitance - Calculatin (cnt.) Electic field lines iginate fm the psitive chages n the suface f ne cnduct and teminate n the negative chages n the suface f the the cnduct. The vltage between the tw cnducts is V. The capacitance f this capacit can be witten as C V Capacitance can be detemined by () assuming a V and detemining in tems f V, () assuming and detemining V in tems f.
Capacitance - Calculatin (cnt.) The cicuital symbl f a capacit is When tw capacits f capacitances C and C ae cnnected in paallel, the ttal capacitance is C p C C C C Cp C Capacitance - Example: Paallel Plate Capacit A paallel plate capacit cnsists f tw paallel cnducting plates f aea S sepaated by a unifm distance d. The space between the plates is filled with a dielectic f a cnstant pemittivity ε. Detemine the capacitance. Slutin: It is bvius that the apppiate cdinate system t use is the Catesian cdinate system. When tw capacits f capacitances C and C ae cnnected in seies, we have C C C s C C C s Capacitance - Slutin: Paallel Plate Capacit Then, we put chages and n the uppe and lwe cnducting plates, espectively. The chages ae assumed t be unifmly distibuted ve the cnducting plates with suface densities ρ s and ρ s, whee ρ s /S. As discussed befe, E 0 inside a cnduct and nmal t the cnduct suface. Applying the Gauss s law, we have D a y ρ s Ea y ρ s ε in the dielectic between the cnducting plates if the finging f the electic field at the edges f the plates is neglected. The ptential diffeence between the tw plates can be calculated as y d y d ρs d V E dl ( ay ρs ε) ( aydy) d ε εs y 0 y 0 S Theefe, C ε which is independent f and V d. Capacitance - Excample Excample: : Spheical Capacit A spheical capacit cnsists f tw cnducting sphees f an inne adius i and an ute adius f. The space between the cnducts is filled with a dielectic with pemittivity ε. Detemine the capacitance. Slutin: Assume chages and n the inne and ute cnducts espectively. Applying Gauss' Law t a spheical suface f adius, ( i < < ), we find E n 4π ε whee n is the unit utwad adial vect. The vltage between the i i cnducts V ( d) d E n 4π ε i Theefe, C V i ε i
Electstatic Enegy Since the electic ptential at a pint in an electic field is defined as the wk equied t bing a unit psitive chage fm infinity (ze ptential) t that pint, t bing a chage (slwly s that kinetic enegy and adiatin effects may be neglected) fm infinity against the field f a chage in fee space t a distance, the amunt f wk equied is We V V W ( V V ) e Electstatic Enegy - Fmulatins Extending the pcedue f binging in additinal chages, we have an expessin f the enegy sted in a gup f N chages as N W V e k k k whee V k, the electic ptential at k, is caused by all the the chages and has the fllwing expessin V k N j ( j k ) j jk Electstatic Enegy - Fmulatins (Cnt.) F a cntinuus chage distibutin f density ρ the fmula f the electic enegy becmes W e Vdv ρ V ' whee V is the ptential at the pint whee the clume chage density is ρ and V' is the vlume f the egin whee ρ exists. Electstatic Enegy - Fmulatins (Cnt.) In tems f field quantities, we have W V ' e dv D E Using D εe f a linea medium, we have We dv W dv εe e D ε V ' In tems f capacitance, we have We CV V '
Summay Culmb s Law F F E whee E n k k k k Gauss Theem ψ D da q enclsed whee DεE A Electic Ptential Capacitance Electstatic Enegy V ba b E dl a CV We D Edv CV V '