Phys 332 Electricity & Magnetism Day 13. This Time Using Multi-Pole Expansion some more; especially for continuous charge distributions.

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1 Phys 33 Electcty & Magetsm Day 3 Mo. /7 Wed. /9 Thus / F., / Multpole Expaso (C 7)..-..,.3. E to B; Loetz Foce Law: felds ad foces (C 7) 5..3 Loetz Foce Law: cuets HW4 Mateals Aoucemets A wod of explaato egadg Wedesday s boad eadg: Ofte the Magetc ad electc teactos ae toduced qute sepaately fom each othe ad t s oly late chaptes (whch may folk eve get to) that you beg to see how they e eally aspects of a sgle electomagetc teacto, aspects that we ve boke apat moe out of hstoc goace ad mathematcal coveece tha a fudametal coceptual dstcto. So the Chapte eadg helps us to te electc ad magetc togethe ad so make a smooth tasto ad buld the pope assocatos metal famewok. Last Tmes We met the oto ad tools of a mult-pole expaso ad bega usg fdg the fst few tems fo dscete chage dstbutos. The mult-pole expaso s much lke a Taylo sees o Bomal sees expaso a way of makg good eough appoxmatos. I ths case, we e essetally expadg tems of the ato of og-to-obseve-locato/og-chage-soucelocato. So, f we put ou og ea / the chage souce ad the souce s small compaed to the dstace fom t to the obseve, the the fst few tems the expaso sees ae qute lkely good-eough. Ths Tme Usg Mult-Pole Expaso some moe; especally fo cotuous chage dstbutos. Summay Multpole Expaso Aga, So P cos Whee P s the th Legede polyomal. P P u u P u 3u

2 Phys 33 Electcty & Magetsm Day 3 V ( ) 4 o q 4 o q P ( ) cos P cos 4 o q O fo a cotuous dstbuto, we e lookg at the tegal ( ) V ( ) P cos d 4 o Wag elatve to the og. If you e qute fa fom the souces, you say, t s lke a pot chage way out thee; f you get a lttle close, you ca see that thee s some slght polazato a lttle moe chage o ths ed tha the othe so t s lke a pot chage + a dpole; you get a lttle close ad you a esolve t s lke a pot chage + a dpole + a quadupole, Just lke the fst few tems of a Taylo Sees Expaso ae gaphcally smple buldg blocks, the fst few tems of a Multpole Sees Expaso ae lke (dffeetal foms of) easoably smple chage dstbutos: The moopole, the Dpole, the Quadupole, the Octopole. Moopole: fa eough fom a (o-eutal) chage dstbuto, the voltage looks lke that of a pot chage whee the total chage s V 4 Q, Q q d. We saw may examples of the electc feld gog to the lmt of what t would be fo all of the chage teated lke a pot chage. What f Q =? A smple example s the physcal dpole equal ad opposte chages (q) sepaated by a dstace d. I that case, V 4 qdcos. Dffeet aagemets of chages have potetals that fall off moe quckly as the dstace gets lage. V V mo V dp V quad The fst two tems ae: V mo Q 4 4 V ˆ ˆ dp cos q q q whee the dpole momet s defed usg the popety ˆ cos : q p ˆ

3 Phys 33 Electcty & Magetsm Day 3 p q. Wag: These posto vectos ae elatve to a og. Just lke a momet of eta exactly what you get depeds o the pot you measug agast ( that case, the axs of otato.) Fo the sees to covege the fastest, you wat the og to be the cete of chage, so all s ae as small as they ca be. Examples/Execses: Poblem 3.3(c) (EXERCISE/EXAMPLE): Fd the fst two tems the multpole expaso fo the fgue show below. The total chage s The dpole momet s Q q, so V mo q 4. p q q 3q aˆ y 3qa y ˆ. The challege s to fd y ˆ ˆ, whch s the pojecto of ˆ (see the dagam o the ght) the y ˆ decto. The pojecto the xy plae gves s ad the pojecto oto the y axs gves s, so y ˆ ˆ ss. The dpole tem fo the potetal s V dp 4 p ˆ 3qa 4 y ˆ ˆ 3qa s s. 4 Example: Suppose a th od of legth L les o the z axs ad s ceteed o z =. If the chage pe legth of the od s z z L 3, what ae the fst two tems the multpole expaso? Dvded the od to small segmets of legth dz lke the oe show below. 3

4 Phys 33 Electcty & Magetsm Day 3 dz z L 3 The chage of a segmet of the od betwee z ad (z + dz) s dz z L dz chage of the od s 4 Q L L / 3 L / 8z x dz 3 L 4 L L.. The total We could also get ths esult by otcg that the chage dstbuto s atsymmetc about z =. The moopole tem s V mo. The dpole momet must be the z decto, so p p zˆ z The dpole momet of oe segmet s q z ˆ z. dq z z L dzz. Itegate ove the legth of the od to fd the total dpole momet: p z dp z L L / L / 4 L L / 5 8 z z dz 3 L 5 The dpole tem s pˆ L Vdp 4 4 L / L zˆ ˆ 4 cos Of couse, wth a appoxmate expesso fo the potetal had, we ca fd the coespodg appoxmate expesso fo the feld. (see ppt.) Electc Feld of a Dpole If the dpole momet pots the z decto ( p pˆ z ), the the potetal s V dp, p ˆ pcos 4 4. If thee ae o othe ozeo multpole momets, the electc feld s p

5 Phys 33 Electcty & Magetsm Day 3 E V dp V ˆ V ˆ V ˆ s p 4 3 cos ˆ s ˆ Execse. Poblem 3.38: Suppose the chage desty o the suface of a sphee of adus R s kcos sphecal coodates ( s the agle fom the z axs). What s the appoxmate electc potetal fa away? Thee ae opposte chages o the top ad bottom halves, so Q =. By symmety, the dpole momet must be the z decto, so p p zˆ z q z ˆ z. The chage o a th g betwee the agles ad d s at the same value of (subtly, detfyg ths s key, because, z-hat s a Catesa coodate, so t s a costat as we go about tegatg ove all the pot chages.) z. Rs R d z Rcos d The aea of the g show above s R d R s d, so ts dpole momet s Add (tegate) up the cotbutos fom gs at all agles () to get dp z q z Make the chage of vaables R s d kcos Rcos. p z kr 3 cos s d. u p z kr 3 3 u du kr 3 4R3 k. 3 3 The total chage of the sphee s zeo, so fa away the appoxmate potetal s u cos, so du s d ad 5

6 Phys 33 Electcty & Magetsm Day 3 V dp 4 p ˆ 4R 3 k z ˆ ˆ kr3 cos Ths s the same as the exact aswe (Eq. 3.87) fo > R, so all of the hghe multpoles ae zeo. Og of the Coodates fo Multpole Expaso Suppose the og of the coodate system s shfted by a vecto a (as show below). QuckTme ad a TIFF (Ucompessed) decompesso ae eeded to see ths pctue. The moopole momet does ot chage, sce the total chage Q s uchaged. The dpole momet the ew coodate system s p d a d d a d p Qa If Q =, the p p. Howeve, f Q, the the dpole momet does deped o the choce of the og. QuckTme ad a TIFF (Ucompessed) decompesso ae eeded to see ths pctue. 6

7 Phys 33 Electcty & Magetsm Day 3 Execse: A sold sphee of adus R has a chage desty of R s. a. Note: sphecal coodates us oly fom to ; the othe half of space s coveed by ug fom to. Make a sketch of the chage dstbuto. What s the sg of the chage dffeet egos? R R R R s z b. Fd the fst two tems the multpole expaso of the electc potetal. Because the dstbuto about the xy axs s atsymmetc, Q = ad V mo. The dstbuto s symmetc about the z axs, so the dpole momet oly has a z compoet: p p zˆ z q z ˆ z The sphee must be dvded to segmets that have the same z compoet. We also wat to be able to calculate the chage of each segmet easly. Sce the chage desty depeds o ad, use th gs betwee the ad ad + d ad betwee the agles ad + d. The volume of such a g s s dd s d d s d z cos d 7

8 Phys 33 Electcty & Magetsm Day 3 The z compoet of the g s posto s momet s z cos, so the z compoet of ts dpole dp z R dq z s s d d cos R d s s cos d R d s s d R d s d R d s d R d s d Itegate ove fom to R ad fom to to get the dpole momet fo the whole sphee. The two tegals ca be doe sepaately. Usg the elato R R d s scos, 4 4 R 3 R R 4 3 R4 3 6 s s cos d s cos d 4 s These gve ( the postve z decto): ad p z V dp 4 R 4 dp z 6 4 R 4 sphee p ˆ R 4 z ˆ ˆ R 4 cos 4 48 Pevew Fo Wedesday, you ll ead about magetc foces. We ll talk about the poducto of magetc foces afte that. 8

9 Phys 33 Electcty & Magetsm Day 3 "Ca we do a example poblem fdg the electc feld of a dpole dffeet coodates ad cofguatos?" Jessca Hde esposes Post a espose Adm Maybe a easy oe pola coodates to stat out wth? Casey P, AHoN swag 4 lphe Ad the maybe a moe challegg oe. Spece ad the a REALLY challegg oe. Rachael Hach "It seems that the dea of a dpole momet s athe mpotat. How ae we gog to use ths ad apply t the futue? Ad ae quadupoles/hghe ode -pole momets just ot qute as commo as moo- ad dpole momets?" Casey McGath Post a espose Adm "Those feld cotous o fgue 3.37a fo a pue dpole ae wedg me out a lttle bt. Do you thk we could go to a bt moe coceptual detal about what exactly s gog o whe you cofe the dpole to the og?" Rachael Hach Post a espose Adm "I would also lke to see vaous examples dffeet coodate systems, etc." Sam Post a espose Adm Flag as appopate Flag as appopate Flag as appopate Flag as appopate "Ca we do a poblem usg multpole expaso but fdg the potetal ove a volume athe tha pot chages?" Jessca Post a espose Adm 9

10 Phys 33 Electcty & Magetsm Day 3 Dop off potetal, feld: Ca we get a bette explaato of the potetals fall offs, ad how t elates to the pue ad physcal dpoles? How may tems to keep? Yeah...how ae we supposed to kow how may pole tems we should use the appoxmato f moopole ad dpole (alteed) ae ot eough? aothe dscete chage dstbuto example: Ca we do poblem 3.3 fo pactce? Move the og, chage the dpole: ca we go ove how the og shft chages the dpole momet ad do some examples volvg that? coodate-fee epesetato of dpole feld: Please do poblem Electc feld of dpole othe coodates: Is t possble to show how the electc feld of a dpole would look lke Catesa o cyldcal coodates? Beakg up oe to seveal? (summg ove dstbuto): Ths s somethg that I wodeed as well. A sgle lage dpole s oe thg, but splttg a dstbuto up to seveal s whee I lose my udestadg. Cotuous chage dstbuto: Could we see a example of a multpole expaso wth a cotuous chage dstbuto? Aye. My questo at the ed of the secto as well. Would you just fd the "cete of mass", o moe the actual cete of chage?