Lecture 1: Electrostatic Fields

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Laureen Rodgers
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1 Lecture 1: Electrosttic Fields Instructor: Dr. Vhid Nyyeri Contct: Clss web site: ir/nyyeri/courses/bee

2 1.1. Coulomb s Lw Something known from the ncient time (here comes mber): two chrged prticles exert force on ech other Electrosttic (Coulomb s) force: F ur [ N] (.1) 4 1 R 1 36 where 1 nd re chrges, R distnce between prticles, u R the unit-ector F / m the permittiity of free spce In this nottion, negtie force mens ttrction, positie repelling.

3 1.1. Coulomb s Lw (Exmple) Find the mgnitude of the Coulomb force tht exists between n electron nd proton in hydrogen tom. Compre the Coulomb force nd the grittionl force between the two prticles. The two prticles re seprted pproximtely by 1 Ångström 1Å 1-1 m. F C R mm N e p FG G N R F Rtio :.7 1 F C 39 G times This is why chemicl bounds re so strong!

4 1.. Electric (electrosttic) Field Electrosttic field due to the chrge : E F N V u R q 4 R C m (4.1) An lterntie definition : df E dq F F lim q q q q For system of two chrges: q (4.) Wht s wrong with it?

1.3. Superposition For seerl chrges plced t different loctions in spce, the totl electric field t the prticulr loction would be superposition (ector summtion) of indiidul electric

5 1.3. Superposition For seerl chrges plced t different loctions in spce, the totl electric field t the prticulr loction would be superposition (ector summtion) of indiidul electric fields: N (5.1) Etot En n1 ector sum! (Exmple): find the EF t P 1 = +1C, 1 = +C, 3 = -3C E E E E tot, P, P, P, P u u u R R R R1 R R (3u 4 u ) u 3u x y y x 3

6 1.3. Superposition (cont) Volume chrge density: 3 [ C / m ] (6.1) Surfce chrge density: s s [ C / m ] (6.) Liner chrge density: l [ C / m ] l (6.3) 1 if i, number of olumes u ' Rd (6.4) 4 R There is differentil electric field directed rdilly from ech differentil chrges

7 1.3. Superposition (Exmple) Clculte the electric field from finite chrge uniformly distributed long finite line. Liner chrge density: (z ) l The unitector : u R z ' u u z z ' We ssume symmetry long z with respect to the obsertion point. Therefore, it will be chrge element t z for eery chrge element t +z. As result, fields in z direction will cncel ech other: i E z i due tosymmetry

8 1.3. Superposition (Exmple, cont) The rdil component: de de cos de de R Combining (3.4.1) nd (3.6.3), we rrie to: de dz l ' 4 ( z ' ) which, combined with (3.8.1) nd integrted leds to: z ' E 1 l l dz ' 4 z ' 3 E l (8.1)

9 1.3. Superposition (Exmple ) Clculte the electric field from n infinite plne chrged with s nd consisting of n infinite number of prllel chrged lines. Symmetry leds to cncelltion of tngent components. Utilize (8.1) nd tht. The liner chrge density: l sdx R x y s y s y s 1 x s y cos tn x' y x y x y y E de dx dx dx

10 1.4. Guss s Lw A chrge is uniformly distributed within sphere of rdius. We cn ssume first tht the chrge is locted t the center. Thn, by (4.1): E u r (1.1) 4 By eluting surfce integrls of both sides E ds 4 ur ds (1.) At the surfce of the sphere, the unit-ector ssocited with the differentil surfce re ds points in the rdil direction. Therefore, ur ur 1 nd the closed surfce integrl is 4

11 1.4. Guss s Lw (cont) Therefore, the integrl form: E ds encl (11.1) Here encl is the chrge enclosed within the closed surfce. By using diergence theorem nd olume chrge density concept: s E ds Ed d E Differentil form: (11.)

12 1.4. Guss s Lw (cont ) For the chrge uniformly distributed within the sphericl olume 3 4 / 3 The olume chrge density: d The totl chrge encl enclosed: ) Outside the sphere: r >, encl = E r d E ds 4 Guss Lw s r Er (1.1) 4 r

13 1.4. Guss s Lw (cont 3) ) Inside the sphere: r < d 3 enc 1 4 r r Guss Lw d 3 r E ds 4 r Er s 3 E r r 3 4 (13.1)

14 1.5. Gussin Surfce A Gussin surfce is closed two-dimensionl surfce through which flux or electric field is clculted. The surfce is used in conjunction with Guss's lw ( consequence of the diergence theorem), llowing to clculte the totl enclosed electric chrge by exploiting symmetry while performing surfce integrl. Commonly used re: ) Sphericl surfce for A point chrge; A uniformly distributed sphericl shell of chrge; Other chrge distribution with sphericl symmetry b) Cylindricl surfce for A long, stright wire with uniformly distributed chrge; Any long, stright cylinder or cylindricl shell with uniform chrge distribution.

15 1.6. Potentil Energy nd Electric Potentil A chrged prticle will gin certin mount of potentil energy s the prticle is moed ginst n electric field. b W F dl E dl[ J] e b (15.1)

16 1.6. Potentil Energy nd Electric Potentil (cont) Imginry experiment: compute totl work required to bring three chrged prticles from - to the shded region. No electric field exists t - nd there re no friction, no grity, nd no other forces. I There re no forces here, therefore, no work is required! W 1 = ;

17 1.6. Potentil Energy nd Electric Potentil (cont ) II x b x We need to oercome the Coulomb s force, therefore, some work is required. x b W dx V ( x x) 4 x x (17.1) since both chrges re positie V 1 is n bsolute electric potentil cused by the chrge 1 V x b dx x x x x ( ) 4 (17.)

18 1.6. Potentil Energy nd Electric Potentil (cont 3) III We need gin to oercome the Coulomb s force, therefore, some work is required. x3 x ( x x1 ) 4 ( x x) 4 x3 x1 4 x3 x W dx dx V V W W W W V ( V V ) Totlly, for the three prticles: tot Note: V V i ji j ji i j 4 x x i j

19 1.6. Potentil Energy nd Electric Potentil (cont 4) Or, for N prticles: W tot N N 1 i j 4 x i1 j1, ji ij i1 Here x ij is the distnce between chrges i nd j; The electrosttic energy cn lso be eluted s 1 We Vd [ J] Note: the totl work in our cse is equl to the totl electrosttic energy stored in the shded region. 1 N V V i i i N 4 j1, ji ij i x (19.1) (19.) (19.3) Note: the totl work (nd the totl energy) do not depend on the order, in which prticles re brought.

20 1.6. Potentil Energy nd Electric Potentil (cont 5) Electric potentil difference between points nd b is the work required to moe the chrge from point to point b diided by tht chrge. We cn express the electric potentil difference or oltge s: b b 1 1 Vb F dl E dl b J b C V E dl [ V ] (.1) V Vb E dl b E dl V V V b b (.)

21 1.6. Potentil Energy nd Electric Potentil (Exmple) Elute the work (chrge times potentil difference) required to moe chrge q from rdius b to rdius. The electric field is E 4 r u r The potentil difference between the two sphericl surfces is b 1 1 V u u dr r r b b r r b (1.1) 1 5 The potentil t r = is ssumed to be nd is clled ground potentil. The electric potentil defined with respect to the ground potentil is clled n bsolute potentil.

22 1.6. Potentil Energy nd Electric Potentil (Exmple, cont) Considering the pth , we notice tht there re only potentil differences while going 1 nd 3 4. Therefore, these re the only pths where some work is required. When moing 3, the potentil is constnt, therefore no work is required. A surfce tht hs the sme potentil is clled n equipotentil surfce. If the seprtion between two equipotentil surfces nd the oltge between them re smll: V V V dv E dl Exdx Eydy Ezdz dx dy dz x y z The electric field V V V V m : E ux uy uz x y z (.1)

23 1.6. Potentil Energy nd Electric Potentil (cont 6) We cn modify eqution (.1) s following: E V (3.1) Note tht when chrged prticle is moed long closed contour, no work is required E dl dv E dl E ds E Electrosttic field is consertie nd irrottionl. s (3.)

24 1.6. Potentil Energy nd Electric Potentil (cont 7) S V ince E V V m Poisson s eqution, Lplce s eqn. when = 1 ( x', y', z ') V( x, y, z) dx ' dy ' dz ' 4 ' ' ' x x y y z z (4.1) The potentil energy would be 1 1 ( ) W e Vd E Vd VE E V d VE ds E Vd W e E Ed E d for R (4.)

25 1.7. Dielectric mterils A mteril cn be considered s collection of rndomly (in generl) oriented smll electric dipoles. If n externl electric field is pplied, the dipoles my orient themseles.

26 1.7. Dielectric mterils (cont) We my suggest tht n externl electric field cuses thin lyer of chrge of the opposite sign t either edge of the mteril. This chrge is clled polriztion chrge. The density of the polriztion chrge: p P (6.1) where P is the polriztion field: P N 1 lim pj j1 (6.) Here p j = du d is the dipole moment of indiidul dipole, N number of toms (dipoles)

27 1.7. Dielectric mterils (cont ) Let us dd the polriztion chrge density to the rel chrge density. The Guss s Lw will tke form: which leds to where p E D C D E P electric ( displcemen t) fluxdensity m (7.1) (7.) (7.3) The totl flux tht psses through the surfce D ds[ C] e (7.4) s

28 1.7. Dielectric mterils (cont 3) Integrting (7.) oer olume, leds to s D ds enc (8.1) Dielectric mterils re susceptible to polriztion. Usully, polriztion is linerly proportionl to the pplied (smll) electric field. Then P ee (8.) where e is the electric susceptibility D (1 ) E E E e r (8.3) r is the reltie dielectric constnt for liner nd isotropic mterils We consider only liner mterils here.

29 1.8. Cpcitnce C V [ F] (9.1) A prllel-plte cpcitor Are A w z (9.) Assume A >> d

30 1.8. Cpcitnce (cont) s enc s A s E ds EA E between pltes b b V E dl Ed C A s, orin cse of dielectric C V s d d Stored energy: A V We E d Ad d A d CV (3.1) (3.) (3.3) (3.4) Assumed uniform field in the cpcitor nd uniform distribution of chrge on pltes

31 1.8. Cpcitnce (Exmple) Clculte the mutul cpcitnce of cox cble with dielectric r inside From the Guss s Lw: D ds D L L enc l (31.1) Potentil difference: b b l l b Vb E dl d ln (31.) The totl chrge: L l C V b ll l b ln b ln L (31.3) (31.4)

32 1.9. Electric currents Let s consider wire J E E drift m (3.1) drift where is the electron olume chrge density, is n erge electron drift elocity, is the mobility of the mteril. m m 1/ R the conductiity (3.) The totl current tht psses through the wire I J ds A (3.3)

33 1.9. Electric currents (cont) In our cse f the current is distributed uniformly in cylindricl wire: I J (33.1) The power density (density of power dissipting within conductor): p J EW m The totl power bsorbed within the olume: P pd[ W] (33.) (33.3)

34 1.9. Electric currents (Exmples) ) Clculte the current flowing through the wire of rdius ; current density: J I u I I uz ddu z I 3 z b) Clculte the power dissipted within resistor with uniform conductiity. The oltge cross the resistor is V, current pssing through is I. P I V L J Ed dddz VI (34.1) L z

CAPACITORS AND DIELECTRICS

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