Fri. 10/23 (C14) Linear Dielectrics (read rest at your discretion) Mon. (C 17) , E to B; Lorentz Force Law: fields

Transcription

1 Fi. 0/23 (C4) 4.4. Linea ielectics (ead est at yu discetin) Mn. (C 7) , 2.3. t B; Lentz Fce Law: fields Wed. and fces Thus. (C 7) 5..3 Lentz Fce Law: cuents Fi. (C 7) 5.2 Bit-Savat Law HW6

2 Plaizatin & lectic isplacement Q d xample da P Cnside a huge sla f dielectic mateial initially with unifm field, and cespnding unifm plaizatin and electic displacement P. P Yu cut a small spheical hle ut f it. What is the field in its cente in tems f and P? F illustative pupses nly, take the plaizatin t e antipaallel t the field, and imagine th t e in the z diectin. By Supepsitin Pinciple, cutting ut a sphee is the same as inseting a sphee f ppsite plaizatin. Quting xample 4.2 (which in tun uilds n 3.9), the field inside a unifmly plaized sphee is sphee Psphee 3 S, we add in a sphee f plaizatin P Adding field added P 3 in. sphee 3 P0

3 Plaizatin & lectic isplacement Q d xample da P Cnside a huge sla f dielectic mateial initially with unifm field, and cespnding unifm plaizatin and electic displacement P. P Yu cut a small spheical hle ut f it. What is the field in its cente in tems f and P? sphee P 3 in. sphee 3 P0 sphee What is the electic displacement in its cente in tems f and P? P sphee shee sphee Thee is n mateial in the sphee, s sphee 0 3 P s P P sphee whee P 2 P

4 dp P d Plaizatin & lectic isplacement Paˆ and P P xecise: Cnside a huge sla f dielectic mateial initially with unifm field, and cespnding unifm plaizatin and electic displacement P. Yu cut ut a wafe-shaped cavity pependicula t P. What is the field in its cente in tems f and P? P Hint: Think f inseting the apppiate wave-sized capacit. What is the electic displacement in its cente in tems f and P? Q d da

5 encl da da tp Q encl tp Bunday Cnditins lectic field, acss chaged suface ttm da ttm Send side height / aea t 0 tp da tp tp da ttm ttm da tp A ttm A( ) Q encl ttm A da tp da Bttm da da suface suface tp ttm

6 tp Bunday Cnditins lectic isplacement, acss chaged suface. encl da Q. encl da tp ttm da ttm Send side height / aea t 0 tp da tp da ttm da Q ttm. encl da tp da Bttm da da suface suface tp ttm tp A ttm A( ) A tp ttm

7 dl tp Bunday Cnditins (static) lectic field, alng chaged suface 0 dl tp Send side height t 0 tp 0 dl tp ttm ttm dl dl ttm tpl ttml( ) ttm 0 0 dl 0 dl tp dl ttm tp ttm tp ttm 0

8 Bunday Cnditins (static) lectic displacement, alng chaged suface P dl tp dl tp P dl Send side height t 0 tp dl tp ttm ttm dl dl ttm ttm dl tpl ttml( ) P tpl P ttml( ) P tp dl tp tp ttm P tp P ttm P P tp ttm dl dl tp ttm dl tp P dl ttm ttm dl ttm P tp ttm dl

9 Alng tp ttm tp ttm P tp P ttm Bunday Cnditins lectic and isplacement fields 0 (culd have guessed as much fm P.) Acss tp ttm tp ttm tp tp aˆ aˆ ttm ttm da tp da Bttm aˆ dl tp dl ttm (culd have guessed as much fm P and P aˆ.) aˆ â ttm tp tp ttm

10 Alng tp Bunday Cnditins lectic and isplacement fields ttm 0 tp ttm P tp P ttm P P aˆ xecise Acss tp ttm tp ttm Ba lectet (like an electic a magnet): unifm P alng axis Sketch P,, and as t ey the unday cnditins. P Only suface chage is the und suface chage n tw faces Thee is n suface chage, s n discntinuity in. P

11 Recall: Atm s Respnse t lectic Field Atm n a stick s F small stetch, fist tem in Tayl Seies (Hk s Law) F int F s int s0 s... extenal F extenal = q extenal F intenal = F extenal F q ext ext F F int F F int int int Fint qs F p int F p int F p int s0 s0 s0 s0 qs... p... p... p... lectic iple mment S, f small enugh stetch, weak enugh field p q ext p ext plaizaility

12 Linea ielectics Pint alng field Chunk f induced diples Linealy pptinal f individual induced diple F chunk f induced diples p Plaizatin = iple density veyne s field ut its wn s dp P d d P d d d efine electic susceptiility t e the pptinality cnstant (and pvide cnvenient fact f.) d e d P 0 e Always linea dielectic P e e O, in tems f plaizatin e P P e e Pemittivity (f nt-s- space) 0 e ielectic Cnstant e

13 Linea ielectics Chunk f induced diples xample: cnside a simplified vesin f plem 4.8. Say we have nly ne dielectic mateial, f cnstant etween tw capacit plates distance a apat. V a ˆ n tp zˆ P a. lectic isplacement,. Gaussian x da utside 0 inside tp inside A Q ttm. encl A ttm inside Q Q. encl xpect nly pependicula t suface and nly inside capacit A A Q ttm. encl. encl 0. lectic Field,. z ˆ zˆ c. Plaizatin, P. P z ˆ ˆ z z ˆ ẑ inside utside ˆ d. Ptential iffeence acss plates, V. tp tp V dl zˆ dz ttm ttm n ttm a e. Bund chage, and. P 0 P nˆ tp zˆ zˆ tp P nˆ P nˆ zˆ zˆ ttm ttm f. fm chage distiutin. Q da encl Q f. encl Q inside Attm inside f f f f zˆ. encl

14 Linea ielectics xample: Altenate / iteative pespective n field in dielectic. Cnside again a simple capacit with dielectic. We ll find the electic field in tems f what it wuld have een withut the dielectic. We ll d this iteatively and uild a seies slutins.. P P 2 0. Say we stat with n dielectic. Initially thee s the field simply due t the chage;. We inset the dielectic and that field induces a plaizatin, P e and the assciated suface chages cntiute a field f thei wn,. zˆ whee P nˆ. 0 s in the ppsite diectin.. This field induces a little cunte plaizatin, P 0 e 0 2 e 0 Which is means a suface chage and esulting field cntiutin f its wn P 0 0 e 0 e n0 2 e n e... As lng as e <, this cnveges t inside e 0 0 Same esult as we gt peviusly 2 P 2. See a patten? 0 e 2 0

15 Linea ielectics xecise: Ty it f yu self. A sphee made f linea dielectic mateial is placed in an thewise unifm electic field. Find the electic field inside the sphee in tems f the mateial s dielectic cnstant,. Yu can take it as a given that a sphee f unifm plaizatin cntiutes field P 3 0

16 xample: A caxial cale cnsists f a cppe wie f adius a suunded y a cncentic cppe tue f inne adius c. The space etween is patially filled (fm t c) with mateial f dielectic cnstant as shwn elw. Find the capacitance pe length f the cale. F the sake f easning this ut, say thee s chage Q unifmly distiuted alng the suface f the cental wie. Q C V V c dl a Gaussian cylinde f sme adius a<s<c. a da Q f. encl 2sL Q Q 2sL Q sˆ a s 2sL Q sˆ s c 2sL s s c C L V V V ln a ds a L c dl c a c Q Q 2sL 2sL Q 2L a 2 ln ln c a a dl ln c c dl ds

17 xecise: Thee ae tw metal spheical shells with adii R and 3R. Thee is mateial with a dielectic cnstant = 3/2 etween adii R and 2R. What is the capacitance? 3R 2R R

18 Fi. 0/23 (C4) 4.4. Linea ielectics (ead est at yu discetin) Mn. (C 7) , 2.3. t B; Lentz Fce Law: fields Wed. and fces Thus. (C 7) 5..3 Lentz Fce Law: cuents Fi. (C 7) 5.2 Bit-Savat Law HW6