4-4 E-field Calculations using Coulomb s Law

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1 1/11/5 ection_4_4_e-field_clcultion_uing_coulomb_lw_empty.doc 1/1 4-4 E-field Clcultion uing Coulomb Lw Reding Aignment: pp Specificlly: 1. HO: The Uniform, Infinite Line Chrge. HO: The Uniform Dik of Chrge. HO: An Infinite Chrge Plne

2 1/11/5 The Uniform Infinite Line Chrge.doc 1/5 The Uniform, Infinite Line Chrge Conider n infinite line of chrge lying long the z-i. The chrge denity long thi line i contnt vlue of C/m. Q: Wht electric field E ( r i produced by thi chrge ditribution? A: Apply Coulomb Lw! We know tht for line chrge ditribution tht: E ( r C ( - r r r r -r d r r r r E ( r

3 1/11/5 The Uniform Infinite Line Chrge.doc /5 Q: Yike! How do we evlute thi integrl? A: Don t pnic! You know how to evlute thi integrl. Let brek up the proce into mller tep. Step 1: Determine d The differentil element d i jut the mgnitude of the differentil line element we tudied in chpter (i.e., d d. A reult, we cn eily integrte over ny of the even contour we dicued in chpter. The contour in thi problem i one of thoe! It i line prllel to the z-i, defined nd y. A reult, we ue for d : d dz dz ˆz Step : Determine the limit of integrtion Thi i ey! The line chrge i infinite. Therefore, we integrte from z to z. Step : Determine the vector r -r. Since for ll chrge nd y, we find: ( ˆ ˆ ˆ ( ˆ ˆ ˆ y z y z ( ˆ ˆ ˆ ˆ yy zz z z r -r + y + z + y + z + + ( ˆ + yˆ + z z ˆ y z

4 1/11/5 The Uniform Infinite Line Chrge.doc /5 Step 4: Determine the clr r -r Since r -r y ( z z + +, we find: Step 5: Time to integrte! E ( r C ( r -r + y + z -z ( r r -r r -r 1 ˆ + ( ˆ ˆ + yy z ( z z ( ( ˆ ˆ ˆ + yy + z z z dz + y + ( z z ˆ ˆ ( ˆ + yy + z z z dz + y + ( z z d + + y z z ( ˆ ˆ + yy ( ˆ ˆ + yy dz ( y z z + + dz + + y πε + y

5 1/11/5 The Uniform Infinite Line Chrge.doc 4/5 Thi reult, however, i bet epreed in cylindricl coordinte: ˆ + yˆ coφ ˆ + inφ ˆ + y coφ ˆ in ˆ + φy y y And with cylindricl be vector: coφ ˆ ˆ + inφ y 1 ( co φ ˆ ˆ in ˆ ˆ ˆ + φy 1 + ( co φ ˆ ˆ ˆ ˆ ˆ φ + in φy φ φ 1 + ( co φ ˆ ˆ in ˆ ˆ ˆ z + φy z z 1 co + in ( φ φ ˆ 1 + -co in + in co ( φ φ φ φ 1 + co + in ˆ ( φ( φ( ˆ z ˆ φ

6 1/11/5 The Uniform Infinite Line Chrge.doc 5/5 A reult, we cn write the electric field produced by n infinite line chrge with contnt denity : ( r πε E ˆ Note wht thi men. Recll unit vector â i the direction tht point wy from the z-i. In other word, the electric field produced by the uniform line chrge point wy from the line chrge, jut like the electric field produced by point chrge likewie point wy from the chrge. It i pprent tht the electric field in the ttic ce pper to diverge from the loction of the chrge. And, thi i ectly wht Mwell eqution (Gu Lw y will hppen! i.e.,: ( r E v ( r ε Note the mgnitude of the electric field i proportionl to 1, therefore the electric field diminihe we get further from the line chrge. Note however, the electric field doe not diminih quickly tht generted by point chrge. Recll in tht ce, the mgnitude of the electric field diminihe 1 r.

7 1/11/5 The Uniform Dik of Chrge.doc 1/5 The Uniform Dik of Chrge Conider dik rdiu, centered t the origin, nd lying entirely on the z plne. r E( r r r r Thi dik contin urfce chrge, with denity of C/m. Thi denity i uniform cro the dik. Let find the electric field generted by thi chrge dik! From Coulomb Lw, we know: E ( r S ( - r r r r -r d

8 1/11/5 The Uniform Dik of Chrge.doc /5 Step 1: Determine d Thi dik cn be decribed by the eqution z. Tht i, every point on the dik h cordinte vlue z tht i equl to zero. Thi i one of the urfce we emined in chpter. The differentil urfce element for tht urfce, you recll, i: d d d d φ z Step : Determine the limit of integrtion. Note over the urfce of the dik, chnge from to rdiu, nd φ chnge from to π. Therefore: < < < φ < π Step : Determine vector r-r. We know tht z for ll chrge, therefore we cn write: ( ˆ ˆ ˆ ( ˆ ˆ ˆ y z y z ( ˆ ˆ ˆ ( ˆ ˆ yy zz y y r -r + y + z + y + z ( - ( ˆ + y y ˆ + zˆ y z Since the primed coordinte in d re epreed in cylindricl coordinte, we convert the coordinte to get:

9 1/11/5 The Uniform Dik of Chrge.doc /5 ( ˆ ˆ ˆ ( ˆ ˆ y z y r r + y + z + y ( ' ˆ ( ˆ ˆ y y y z z ( coφ ( inφ + + ˆ + y ˆ + z ˆ y z Step 4: Determine r -r We find tht: ( - coφ ( inφ r -r + y + z Step 5: Time to integrte! E ( r S ( r r -r r -r π d ( - coφ ( inφ ˆ + y ˆ + z ˆ y z ( - coφ ( inφ + y + z d dφ Yike! Wht me! To implify our integrtion let determine E r long the z-i only. In other word, the electric field ( et nd y.

10 1/11/5 The Uniform Dik of Chrge.doc 4/5 ( r r -r E(, y, z d S r -r π ( coφ ˆ ( ˆ ˆ + inφ y z z 4 d dφ πε ( coφ + ( inφ + z π ( coφ ˆ ( in ˆ ˆ + φ y z z d d 4 φ πε + z π ( coφ d d φ ˆ + z π ( inφ d dφ + ˆ y + z π z d dφ + ˆ z + z Note tht ince: π π inφ dφ coφdφ The firt two term (E nd E y re equl to zero. Integrting the lt term, we get: E (,y,z z ˆ z 1 if z > ε z + z ˆ z 1 if z < ε z +

11 1/11/5 The Uniform Dik of Chrge.doc 5/5 From thi epreion, we cn conclude two thing. The firt i tht bove the dik (z >, the electric field point in the direction ˆ z, nd below the dik (z <, it point in the direction -ˆ z. Wht urprie (not! The electric field point wy from the chrge. It pper to be diverging from the chrged dik ( predicted by Gu Lw. Likewie, it i evident tht we move further nd further from the dik, the electric field will diminih. In fct, ditnce z goe to infinity, the mgnitude of the electric field pproche zero. Thi of coure i imilr to the point or line chrge; we move n infinite ditnce wy, the electric field diminihe to nothing.

12 1/11/5 An Infinite Chrge Plne.doc 1/ An Infinite Chrge Plne Sy tht we hve very lrge chrge dik. So lrge, in fct, tht it rdiu pproche infinity! Q: Wht electric field i creted by thi infinite plne? A: We lredy know! Jut evlute the chrge dik olution for the ce where the dik rdiu i infinity. In other word: ˆ z z 1 if z > ε z + lime(, y, z ˆ z z 1 if z < ε z + ˆ z if z > ε ˆ z if z < ε Therefore, the electric field produced by n infinite chrge plne, with urfce chrge denity, i:

13 1/11/5 An Infinite Chrge Plne.doc / E ( r ε ˆ if z > z ˆ z if z < ε Think bout wht thi y! * Firt, we note tht the electric field point wy from the plne if i poitive, nd towrd the plne if i negtive. * Second, we notice tht the mgnitude of the electric field i contnt the mgnitude i independent of the ditnce from the infinite plne! >

14 1/11/5 An Infinite Chrge Plne.doc / The reon for thi reult i, tht no mtter how fr you re (i.e., z from the infinite chrge plne, you remin infinitely cloe to plne, when compred to it rdiu. We will find thee reult re ueful when we tudy the behvior of prllel plte cpcitor.

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