ρ θ φ δ δ θ δ φ δ φ π δ φ π δ φ π

Physics 6 Fin Ex Dec. 6, ( pts Fou point chges with chge ± q e nged s in Figue. (5 pts. Wht is the chge density function ρ (, θφ,? (,, q ( ( cos ( / + ( ( / / ρ θ φ δ δ θ δ φ δ φ π δ φ π δ φ π b (5 pts. Wht e the utipoes q (do s ny s you cn; thee is usefu infotion in the HW soutions? In pticu, wht is the fist non vnishing utipoe? ( θ, φ ρ(, θ, φ q Y dd Ω * ( ( + ( ( + ( ( + ( ( + +! q P e + e e π! iπ/ iπ iπ/ +! iπ / q P e + ( π! +! q P ( i + ( π! +! q P π! / fo even P vnishes uness+ is even so ust be even so. The fist non vnishing utipoe needs to hve nd is 5 5 q q q π! π c ( pts. Do utipoes fo odd wys vnish? Cn you expin why o why not using the ottion syety in the souce? Yes. Notice tht ottion bout the z xis by π eves the chge density identic to the ρ, θφ, + π ρ, θφ,. But unde ottion by π initi density, i.e. iπ ( θ, φ + π ( θ, φ ( ( θ, φ Y Y e Y * * * Theefoe, fo ottiony syetic distibutions

* ( θ, φ ρ(, θ, φ ( θ, φ π ρ(, θ, φ π ( ( θ, φ ρ(, θ, φ ( q q Y dd Ω * Y + + dd Ω * Y dd Ω Fo odd, the utipoe oents ust vnish. d (5 pts. Wht is the potenti t octions with > nd <? By the ddition theoe q π Φ + + πε + < * * * * (, θ, φ Y (, Y (, π / Y (, π Y (, π / Y ( θ, φ + > ( q +! ε < P + + > π +! / ( Y ( θφ, whee < nd > e the se nd ge of nd. e (5 pts. Suppose the chges e encosed in gounded conducting sphee of dius b centeed on the oigin. Wht e the potentis fo > nd <? The esiest ethod is to sipy dd to the soution in d the negtive of the soution of Lpce s eqution tht hs the coect boundy potenti vue ( q +! Φ,,, < tot ( θφ P Y + ε + > π +! ( P + + b π +! / ( θφ q +! + ε / ( Y ( θ, φ q! < + / P ( Y( θφ, + + ε + > b π ( +! whee < nd > e the se nd ge of nd. This soution woud so be obtined by dding the potentis of the fou ige chges with stength q qb/ t octions b /.

f ( pts. Fo the soution in e, wht is the chge density on the inside sufce of the sphee? σ ε E ( ( + / ( Y ( θ, φ (! E P Y, Φ q b + ( + + + b ε + b b π +! q +! σ P π! b b / ( θ φ g ( pts. Wht is the fied outside the sphee? Becuse the chge is inside the conducting sphee, by Guss s Lw the fied outside the sphee is shieded wy to zeo. (5 pts. Sove the sque D potenti pobe given by Figue. The foowing steps y be hepfu. (5 pts. Wht e the fundent soutions in Ctesin coodintes? An expe of n expnsion set is ( xy, Φ ( π + ( π + ( π + ( π ( π + ( π + ( π + ( π A sin x/ e B cos x/ e C sin x/ e D cos x/ e πy / πy / πy / πy / E sin y/ e F cos y/ e G sin y/ e H cos y/ e πx / πx / πx / πx / b (5 pts Sove the boundy vue pobe with the potenti Φ t x, < y< nd x, < y < equ to V nd with the othe sides hving Φ. Hee becuse of the boundy condition t y nd y πx (, sin ( π / sin ( π / Φ xy E y e + G y e Becuse of the boundy conditions t x nd x Othogonity gives / πx / ( π ( π V E sin y/ + G sin y/ π ( π ( π V E sin y/ e + G sin y/ e π

V ( cos π + ( E + G π V π π ( cos π + ( Ee + Ge π V π π V π E ( cos π + ( e / ( e ( cos π + ( e / sinh π π π V V G cos π + e / e cos π + e / sinh π π π π π π V π ( π π / Φ xy, sin y/ sinh x/ sinh x π sinh π odd Note tht ust be odd fo non zeo expnsion coefficient. c (5 pts. Sove n ppopite boundy vue pobe to incude the fces t Φ V. The soution is the se switching x y nd evesing the sign of the fce vues: V Φ ( x, y sin ( πx/ sinh πy/ sinh π ( y / π sinh π odd d (5 pts. Wht is the tot potenti? By supeposition tot odd ( xy, ( xy, ( xy, Φ Φ +Φ ( ( ( π π π ( sin y/ sinh x/ sinh x / V π sinh π sin ( πx/ sinh πy/ sinh π ( y / e (5 pts. Put pied coodinte syste with its oigin t x ( /, / the θ fied?. Ne wht is dependence of the potenti? Ne wht is the θ dependence of the eectic (5 pts. A unifoy chged infinitey thin spheic she of chge q nd dius is spun ound the z xis with constnt ngu fequencyω. J, θ, φ is (5 pts. Show tht the cuent density function qω J(, θ, φ δ ( sin θφˆ. π The esy wy to get this is to eize the chge density is

q ρ π ( δ, nd the ottion veocity is v ωsinθφˆ. Mutipying these two esuts gives the expession. Anothe wy is to conside the cuent pssing though the e eeent ddθ t po ngeθ. The tot chge in the ing of chge t this oction is ( q nd / sinθdθ. This chge psses the oction on the sphee t θ with fequency ω /π, so the oc cuent is qω /π sinθdθ. The no to the e eeent is ˆ φ. Now qω qω Jdd φ θ sinθdθ Jφ δ ( sinθ π π b (5 pts. Wht is the vecto potenti in spce fo this cuent souce ssuing the vecto potenti vnishes s? J( x qω δ ( sinθ ˆ A d x φ d x π xx 6π xx ˆ qω δ ( sinθ Aφ A φ [ cosφcosφ + sinφsinφ ] dd Ω 6π x x ( φ( Y ( θ φ Y ( θ φ ( ( cos φ Y ( θ, φ Y ( θ, φ + ( + ( 8π qω δ π dd Ω x x sin φ Y θ, φ Y θ, φ / i 8π qω cos,, < δ ( d 8π > + sin φ Y θ, φ + Y θ, φ / i qωsinθ δ < ( d π > qωsinθ < π Aφ qω sinθ > π c (5 pts. Wht is the gnetic induction inside the she?

ˆ ˆ θ B A ( sinθ Aφ ( Aφ sinθ θ cos ( sin cos ˆ sin sin ˆ cos ˆ qω θ θ φx + θ φy+ θz π sinθ( cosθcosφxˆ+ cosθsinφyˆsinθzˆ qω σω zˆ ˆ z π wheeσ is the (unifo! sufce chge density. d (5 pts. Wht is the gnetic induction outside the she nd the gnetic oent? qω σω cosθ + sinθθ cosθ + sin θθˆ, π B ˆ ˆ ˆ cssic dipoe fied distibution. Coping to Eqn. 5., the oent is coud so be obtined by diect integtion too. qω /. This esut d diπ sin ω ω q π π π di dq d cos d ωq ωq ωq sin θdcosθ [ / ] θ θ φ e (5 pts. Wht is the gnetic enegy inside nd outside the she? The gnetic enegy inside the she is esy B B π σ ω T V 7 5 σω π 8 The gnetic enegy outside the she is ony sighty oe txing.

T B B d x cos + sin π σω θ θ dd cosθ π σω cos θ + dd cosθ σω one hf the enegy inside! d 8π σω π 8, ( pts. In ou fin hoewok set we det with cceeto dipoe gnets. In this pobe, sove sii pobe fo D cceeto qudupoe gnets with ode cuent density NI J z (, θ cos θδ (. ( pts. Wht is the vecto potenti geneted by this souce ssuing the vecto potenti vnishes s? The vecto potenti is A z (, θ ( sin θ cos θ ( sin θ cos θ Continuity nd othogonity t A z (, θ A + B < C + D > ipy ( sin θ cos θ ( sin θ cos θ A + B < A + B > The jup condition t nd othogonity ipy A, B fo, nd

Az Az NIcos θ + ε ε ( ( B B NI B NI Theefoe A z (, θ NI NI cos θ cos θ < > b ( pts. Wht is the gnetic induction inside nd outside? Tking the cu of the vecto potenti B B θ (, θ (, θ NI Az θ NI NI cos θ Az NI cos θ sin θ < sin θ > < > c ( pts. Wite the induction fo < in tes of the Ctesin coodintes x nd y nd the Ctesin unit vectos ˆx nd ŷ. Wht e the powes of x nd/o y tht ppe? B B ˆ ˆ + Bθθ NI sinθcosθ ( cosθxˆ sinθyˆ ( cos θ sin θ( sinθxˆ cosθyˆ + + + NI NI [ sinθxˆ+ cosθyˆ] [ yxˆ+ xyˆ] The fied is ine in the Ctesin vibes. d ( pts. Wht is the gnetic enegy pe unit ength inside nd outside? The gnetic enegy inside

T The gnetic enegy outside B B ddθ NI π d NI N I π π 6 NI T NI π π ( the se s inside (this is esut ike Pobe 5. in Jckson! 5 sin θ + cos θ ddθ N I π 6, e ( pts. Wht is the sef inductnce pe unit ength of the gnet ssuing I is the tot cuent enteing (nd eving! the gnet? LI N I N I T π +π 6 6 N L π 8 N I π 8 Ext cedit (5 pts.: Given wht you edy know, wht is function fo of the potenti s function of ndθ, nd so of x nd y fo sextupoe gnet? (you ve done di(twopoes nd qud(fouupoes edy now!

Figue Figue