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

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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

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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

Phys 309 Leo 9 p 5 6

Solution

Phys 309 Lec 9 p 6 6