Phys309 Lec 9 p 1 6
Today Solution to Laplace5 equation for problems withazimuthalsymmetryby themethodofseparationof variables in sphericalcoordinates
In sphericalcoordinates Laplace equationbecomes usingEg 1.73
o Ev rt Irl r En t pestoFolsin03 trsino34ozForproblems of azimuthalsymmetry rotationalsymmetryaboutthe 2 axisV is independentofof and this Ltimes r2 reduces to
o ZrLriff sinZolsinoffoWe first tookforseparablesolutions r O Bln OThe lasteq dividedbyV becomes
than Rotary t s.inddolsin0daEo o
oftheform f r t g O 0 which is possibleonly if each term is aconstant
thanhidden a osinddolsinodaft c
Theangular equationhas nonsingulan smooth solutionsonlyfor C LLet2 1 942inwhichcase thesolutions are the Legendrepolynomials
O Peloso where
Pelx Fe adz le LE 1 l the Rodriguesformula for Petx 3.62
The first fewLegendrepolynomials are
Note that
Pelx is an lthorderpolynomial in Xeven
Petey is odd for l corded
Normalization convention is Pele 1
In fact there is a 2ndsolution for each l as there must befora2ndorderODEbut it blowsup at 0 0 and or Tl
Given that c lLett the radialeg becomes
ITLr2dR llett R with generalsolution Rln Arlt Fetty
Phys 309 Lec 9 p 2 6
Thus the general separablesolution smoothawayfrom neo is
Vfr 0 Aeret Bente Pelcoso
and the generalsolution is obtained via superposition
Vtr 0 Aeret Bet PeLeos0 13.65
solution
Since the sphere is hollow it is anphysical for V toblowup at n oTherefore Be O and
oLr 0 E Aert PeleosO
f 0
Then since VLr O Vo O at n e R we have
Volo AeRl PeLeos01So the coefficients AeRe are the Legendrecoefficients of the Legendre seriesf or Volo analogous to Fourier coefficientsof a Fourier series
µXecoso
Using theorthogonalityproperty Axe SinOdo
Pelx Pel x dx fo pelcoso pic so si OdoSx i Soka So o
O e e2
e L
we canmultiply the last sum by a particular PeLoos0 and then apply fothsinodoto pickout a particularcoefficient
fo VolOtPeLoos0 sinOdo Lefty AeRe
Ae 2ff SoaVoloPelcoso do
Sometimeswe can find the coefficientsofthe Legendreseries Volo AeRtPetcosobyinspection withoutintegration Forexample for VOLO K cos2 Oz wehave
Voloy Litcos07 KzLpocost t Pilcoso Ao E AnR E Aro fore I
I
Phys 309 Lec 9 Po3 6
Solution Vln01 Ezo Aeret Brett Peloso
As usual we require 0 as r no Therefore Ae 0 and
Vtr O eEoFeetPeloso
At r R we are given that VCRO VOLO so
Vol01 E BeKo Fei
Petcosoy
Multiplyingby PeLoos01 andapplying fo 5inOdd to both sidesSo VololpeilcosolsinOdo 2ft BRETT
Be2k Ret fo'TVoloPelcos01sinOdo
Themostimportantapplication isprobably the followingproblems and its analogfon a dielectric rather than conducting sphere see Example 4,7 of chapters4
Solution Vtr O Aerlt Yet Pelcoso
We are interested in the regionoutside the sphereTheconductingsphere is an equipotential Vo VoPo cos01 for someconstantVo
nTosaythat thesphere is placedin a uniform electric field E EoZ means that
E EOE as n a
EoZ t C as in no for some constant CEorcosO t C Eor Palcoso t C
So the BC's ofthis problem are Lil VoPoLeos01 at n Rwit EonPilcoSO t C as r x
From Bec Lil we have
Aot BRI Vo and AeRet Beget D for l 0
Bo R Vo Ao and Be AeRUH for l O
I e e
Phys 309 Lec 9 p 4 6
The series for Lr 0 becomes
Vlr O VoEn t Ee Ae re 71Pelcos01Then from BC Lili we have
Ao C A Eo and Ae o for E 2 1
This givesLno VoRf T Cli Rf Eo r Rfa cos0
Although we cannotimpose our usual convention 0 as r oo in thisproblem we can at least require that we recover this condition when Eo D
Thisgives CIO and
VLnO VoRf Eo r Ira cos0
InterpretationVoRy is the potential dueto a uniformlychanged
spherical conducting shell
EorcoSO is thepotential due to the external field
to Pipcos0 is the dipolepotentialdue to the redistributionofchangeon the conducting shell inducedby the external field
The surface changedensity is
0101 Eo 21artnerEo VoRpt Eo et277 cos0 lr rEofYog t 3Eocoso
AThis term ispositive in the northernhemisphere OEO Itandnegative in the southernhemisphere L LOST
The second term averages to zero on the sphere so that the total changeQis related to Vo via
Q fgpn.netO da Gang area 11471132 4ThEoVoR
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