Introduction. Electrostatics

Size: px

Start display at page:

Download "Introduction. Electrostatics"

Albert Rogers
6 years ago
Views:

1 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

2 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

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

4 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.

5 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

6 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.

7 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.

8 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

9 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 '

10 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 '

Electric Charge. Electric charge is quantized. Electric charge is conserved

Electric Charge. Electric charge is quantized. Electric charge is conserved lectstatics lectic Chage lectic chage is uantized Chage cmes in incements f the elementay chage e = ne, whee n is an intege, and e =.6 x 0-9 C lectic chage is cnseved Chage (electns) can be mved fm ne