Electric Fields and Their Solutions
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Transcript of Electric Fields and Their Solutions
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7/25/2019 Electric Fields and Their Solutions
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7/25/2019 Electric Fields and Their Solutions
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'SYPSQFW 0E[ 0SVIRX^ *SVGI 0E[
*YRHEQIRXEP 0E[W
TSMRX GLEVKI
'SYPSQFW 0E[
7XYH] +YMHI
)\EQ
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- [ERX XLI FMKKIWX TSWMXMZI GLEVKI SR XLI VMKLX
XS TYWL MX EW LEVH EW TSWWMFPI XS XLI PIJX 0MOI[MWI- [ERX XLI QSWX RIKEXMZI GLEVKI XS FI EW GPSWI EW
TSWWMFPI XS XLI SVMKMREP GLEVKI FYX SR MXW PIJX 8LI
PEWX GLEVKI NYWX LEW XS KS XS XLI PIJX SJ IZIV]XLMRK
WMRGI XLEX MW XLI SRP] TPEGI PIJX
% TSMRX GLEVKI U ! QMGVSGSYPSQF MW TPEGIH EX XLI TSWMXMSR \ ! Q SR XLI \E\MW
8LVII TSMRX GLEVKIW U ! R' U ! R' ERH U ! R' EVI EPWS TPEGIH WITEVEXIP]
SR XLI \E\MW EX XLI XLVII TSWMXMSRW \ ! Q \ ! Q ERH \ ! Q ;-8,398 74)'-*=-2+
;,-', 437-8-32 ;390( '366)7432( 83 ;,-', 3* 8,) 8,6)) ',%6+)7 %VVERKI
XLI XLVII GLEVKIW XS RH XLI QE\MQYQ JSVGI SR XLI TSMRX GLEVKI U XS TSMRX MR XLI2)+%8-:) \ HMVIGXMSR 8LI VIWYPXERX JSVGI MW
I
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I
8[S WQEPP WTLIVIW IEGL [MXL QEWW Q ! K ERH GLEVKI U EVI WYWTIRHIH JVSQ E
SMRX F] XLVIEHW SJ PIRKXL 0 ! Q ;LEX MW XLI GLEVKI SR IEGL WTLIVI MJ XLI XLVIEHW
EOI ERH ERKPI XLIXE ! HIKVIIW [MXL XLI ZIVXMGEP#
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8LI IUYMPMFVMYQ TSMRX JSV XLI IPIGXVSR QYWX PMI
SR XLI PMRI [MXL XLI JSVQYPE
% TSMRX GLEVKI U ! ' MW EX XLI SVMKMR ERH E WIGSRH TSMRX GLEVKI U ! ' MW
X XLI TSMRX \ ! Q ] ! Q *MRH XLI \ ERH ] GSSVHMREXIW SJ XLI TSWMXMSR EX [LMGL
R IPIGXVSR [SYPH FI MR IUYMPMFVMYQ
8LI JSPPS[MRK WLS[W XLEX F MW XLI GSVVIGX ERW[IV
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8LI YTTIV LEPJ SJ E VMRK MW YRMJSVQP] GLEVKIH [MXL E TSWMXMZI GLEVKI HIRWMX] WMKQE [LMPI
XLI PS[IV LEPJ MW YRMJSVQP] GLEVKIH [MXL E GLEVKI HIRWMX] SJ WMKQE ;LEX MW XLI HMVIGXMSR
SJ XLI IPIGXVMG IPH EX XLI TSMRX < SR XLI TIVTIRHMGYPEV E\MW SJ XLI VMRK EW WLS[R#
)PIGXVMG IPHW
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;LMGL SRI SJ XLI JSPPS[MRK KYVIW WLS[W E UYEPMXEXMZIP] EGGYVEXI WOIXGL
SJ XLI IPIGXVMG IPH PMRIW MR ERH EVSYRH XLMW W]WXIQ#
(IWGVMTXMSR
8LMW MW E QIXEP WTLIVI [MXLE QIXEP GSRGIRXVMG WTLIVMGEP
WLIPP
)PIGXVMG IPH PMRIW
8LI RYQFIV SJ IPH PMRIW TEWWMRK XLVSYKL
E KMZIR WYVJEGI MW TVSTSVXMSREP XS XLI GLEVKI
IRGPSWIH F] XLI WYVJEGI
7II )\EQ *EPP JSV ERSXLIV I\EQTPI
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8[S MRRMXI GLEVKIH TPERIW LEZI GLEVKI HIRWMXMIW WMKQE ERH WMKQE -J WMKQE "
[LEX MW XLI IPIGXVMG IPH FIX[IIR XLI TPERIW#
6IKMSR 6IKMSR 6IKMSR
2SXMGI XLEX MR 6IKMSR XLI X[S IPIGXVMG IPHW FSXL TSMRX XS[EVHW XLI VMKLX TSWMXMZI \ WS
QEKRMXYHI SJ IPIGXVMG
IPHW SRP]
4VMRGMTPI SJ 7YTIVTSWMXMSR
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% TVSXSR MW WLSX ZIVXMGEPP] YT[EVH [MXL E ZIPSGMX] Z!I QW NLEX MR E YRMJSVQ
PIGXVMG IPH ) ! 2' NLEX ,S[ JEV HSIW XLI TVSXSR XVEZIP FIJSVI MX MW FVSYKLX
QSQIRXEVMP] XS VIWX# 2IKPIGX KVEZMX]
Q
)PIGXVSH]REQMGW
%PXIVREXMZI HIVMZEXMSR YWMRK GSRWIVZEXMSR SJ IRIVK]
Q
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)PIGXVMG (MTSPIW
9RMJSVQ
HMTSPI
QSQIRX
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%R MWSPEXIH WMRKPI RIYXVEP [EXIV QSPIGYPI ,3 LEW ER IPIGXVMG HMTSPI QSQIRX SJ
QEKRMXYHI I 'Q -J XLI QSPIGYPI MW TPEGIH MR ER IPIGXVMG IPH SJ I 2' [LEX
QE\MQYQ XSVUYI GER XLI IPH I\IVX SR MX#
8LI QE\MQYQ XSVUYI SGGYVW [LIR XLI IPIGXVMG IPH MW TIVTIRHMGYPEV XS XLI HMTSPI QSQI
I 2Q
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%R IPIGXVMG HMTSPI MW SVMIRXIH TEVEPPIP XS E YRMJSVQ IPIGXVMG IPH EW WLS[R -X MW VSXEXIH
S SRI SJ XLI ZI SVMIRXEXMSRW WLS[R FIPS[ 6ERO XLI REP SVMIRXEXMSRW EGGSVHMRK XS XLI
LERKI MR XLI TSXIRXMEP IRIVK] SJ XLI HMTSPIIPH W]WXIQ QSWX RIKEXMZI XS QSWX TSWMXMZI
;LIR XLI IPIGXVMG IPH TSMRXW MR XLI WEQI HMVIGXMSR EW XLI HMTSPI QSQIRX XLIR XLI HMTSPI MW
MR MXW PS[IWX IRIVK] GSRKYVEXMSR 8LMW MW IZMHIRX MR SYV TEVXMGYPEV GSRKYVEXMSR WMRGI XLI
JSVGI SR XLI TSWMXMZI GLEVKI TSMRXW XS XLI VMKLX ERH XLI JSVGI SR XLI RIKEXMZI GLEVKI TSMRXW
XS XLI PIJX
PS[IWX IRIVK]
6ERO JVSQ PS[IWX
TSXIRXMEP IRIVK] XS
LMKLIWX TSXIRXMEP IRIVK]
%
& '
( ERH XMI XLIR
) ERH XMI XLIR
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2IYXVEP EXSQ EW QER] TSWMXMZI GLEVKIW EW RIKEXMZI GLEVKIW
%TTP] ER IPIGXVMG IPH XS XLI EXSQ
)PIGXVSRW EVI EXXVEGXIH F] XLI IPIGXVMG IPH ERH TVSXSRW EVI VITIPPIH F] MX EGGSVHMRK
XS 'SYPSQFW 0E[
8LMW HMWTPEGIQIRX SJ IPIGXVMG GLEVKI XYVRW XLI EXSQ MRXS E WQEPP HMTSPI [MXL ER IPIGXVMG
IPH SJ MXW S[R
8LI RIX IPIGXVMG IPH MW VIHYGIH
F] XLIWI EXSQW IZIR XLSYKL
XLI] EVI RIYXVEP
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LI HMIPIGXVMG MR E GETEGMXSV MW GSQTSWIH SJ PSXW SJ RIYXVEP EXSQW WSQI OMRH SJ MRWYPEXSV
LEX YPXMQEXIP] VIHYGIW XLI WXVIRKXL SJ XLI IPIGXVMG IPH EW HIWGVMFIH SR XLI TVIZMSYW TEKI
%PXLSYKL XLI IPIGXVMG IPH MW WQEPPIV MRWMHI XLI GETEGMXSV [MXL E HMIPIGXVMG XLI
GETEGMXERGI MW PEVKIV FIGEYWI SJ XLI [E] XLI HMIPIGXVMG GSRWXERX OETTE XW
MRXS XLI EPKIFVE EFSZI
HMIPIGXVMG
GSRWXERX
8LI IPIGXVMG IPH ERH TSXIRXMEP HVST EVI FSXL WQEPPIV MR E GETEGMXSV [MXL E HMIPIGXVMG XLE
XLI] EVI MR E GETEGMXSV [MXLSYX E HMIPIGXVMG
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'SYPSQFW 0E[! +EYWW 0E[
+EYWWMERWYVJEGI
7II LXXTL]TIVTL]WMGWTL]EWXVKWYIHYLFEWIIPIGXVMGKEYPE[LXQP JSV PSXW QSVI I\EQ
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'SRWMHIV E WTLIVMGEP +EYWWMER WYVJEGI SJ VEHMYW Q [LMGL WYVVSYRHW X[S IPIGXVMG
MTSPIW ERH X[S GLEVKIW SRI TSWMXMZI ERH SRI RIKEXMZI EW WLS[R MR XLI KYVI ,IVI
! R' U! R' ERH T!T!B_a 'Q ;LEX MW XLI RIX IPIGXVMG Y\ XLVSYKL
LI +EYWWMER WYVJEGI#
6IQIQFIV XLEX E HMTSPI MW GSQTSWIH SJ X[S GLEVKIW SJ IUYEP QEKRMXYHI FYX STTSWMXI
WMKR WS XLI] [MPP RSX GSRXVMFYXI XS XLI RIX GLEVKI IRGPSWIH F] XLI WYVJEGI
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IPIGXVSRW WTVIEH
SYX YRMJSVQP]WTLIVMGEP
W]QQIXV]
WEQI Y\ XLVSYKL IEGL
JEGI SJ XLI GYFI
% QIXEP WTLIVI SJ VEHMYW V MW TPEGIH EX XLI GIRXIV SJ E GYFMGEP FS\ [MXL E WMHI PIRKXL SJ
[LIVI 0 "" V -J I\GIWW IPIGXVSRW EVI TPEGIH SR XLI WTLIVI [LEX MW XLI QEKRMXYHI SJ
LI IPIGXVMG Y\ XLVSYKL SRI JEGI SJ XLI FS\#
,IVI MW E XVMGO] TVSFPIQ 7YTTSWI XLEX E GLEVKI MW EX XLI GSVRIV SJ E GYFI ;LEX MW XLI
Y\ XLVSYKL SRI WMHI SJ XLI GYFI RSX XSYGLMRK XLI GLEVKI#
(VE[ IMKLX MHIRXMGEP GYFIW WYVVSYRHMRK XLI GLEVKI
EW WLS[R MR XLI KYVI FIPS[
XLIVI EVI
WUYEVIW SR
XLI JEGIW SJ
XLMW RI[
+EYWWMER
WYVJEGI
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% WTLIVMGEP QIXEP WLIPP LEW E GLEVKI TIV YRMX EVIE WMKQE ERH E VEHMYW 6 ;LEX MW XLI
QEKRMXYHI SJ XLI IPIGXVMG IPH EX E HMWXERGI \ JVSQ XLI WYVJEGI SJ XLI WTLIVI#
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+EYWWMER WYVJEGI
3RP] XLI WMHIW GSRXVMFYXI XS XLI
MRXIKVEP WMRGI XLI XST ERH FSXXSQ
YRMX ZIGXSVW EVI SVXLSKSREP XS
I
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'SRWIVZEXMSR SJ GLEVKI GLEVKI HSIWRX
WTSRXERISYWP] ETTIEV ERH HMWETTIEV1EXLIQEXMGEPP]
'SRWIVZEXMSR SJ IRIVK]
)\EQTPI
- GSRRIGX X[S GETEGMXSVW XS E FEXXIV] 8LI JVII IPIGXVSRW SR XLI PIJX WMHI SJ ' EVI
EXXVEGXIH XS XLI FEXXIV] ERH HITSWMXIH SR XLI VMKLX WMHI SJ ' 8LI [MVI MR FIX[IIR
XLI GETEGMXSVW MW RIYXVEP FYX XLI JVII IPIGXVSRW SR XLI PIJX WMHI SJ ' EVI VITIPPIH F]XLI IPIGXVSRW SR XLI VMKLX WMHI SJ XLI TPEXI ERH QSZI XS XLI VMKLX WMHI SJ ' ;I EVI
XLIR PIJX [MXL X[S GLEVKIH GETEGMXSVW [MXL IUYEP GLEVKI 'ETEGMXSVW MR WIVMIW LEZI
XLI WEQI GLEVKI
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'ETEGMXSVW MR WIVMIW LEZI XLI WEQI
GLEVKI SR XLIQ
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'SRWIVZEXMSR SJ GLEVKI
2S[ PIX XLI X[S WTLIVIW XSYGL %JXIV XSYGLMRK XLI] GSQI XS XLI WEQI TSXIRXMEP
WMRGI GSRHYGXSVW EVI IUYMTSXIRXMEPW
'SRWIVZEXMSR SJ GLEVKI GSQFMRIH [MXL IPIGXVMG TSXIRXMEP
2S[ [I WITEVEXI XLI GLEVKIW EKEMR ERH I\EQMRI XLIMV IPIGXVMG TSXIRXMEPW
-J XLI WTLIVIW EVI MHIRXMGEP XLIR XLI] [MPP LEZI XLI WEQI GLEVKI EJXIV XSYGLMRK FYX MJ
XLI] EVI RSX MHIRXMGEP XLIR SRI [MPP LEZI QSVI XLER XLI SXLIV HITIRHMRK SR XLI WM^I SJ
MXW VEHMYW
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8[S MHIRXMGEP GSRHYGXMRK WTLIVIW ERH GEVV] IUYEP EQSYRXW SJ GLEVKI ERH EVI \IH
E HMWXERGI ETEVX XLEX MW PEVKI GSQTEVIH [MXL XLIMV HMEQIXIVW 8LI WTLIVIW VITIP IEGL SXLIV
[MXL ER IPIGXVMGEP JSVGI SJ * 7YTTSWI RS[ XLEX E XLMVH MHIRXMGEP WTLIVI LEZMRK ER
MRWYPEXMRK LERHPI ERH MRMXMEPP] YRGLEVKIH MW XSYGLIH VWX XS WTLIVI XLIR XS WTLIVI
ERH REPP] VIQSZIH *MRH XLI JSVGI FIX[IIR WTLIVIW ERH RS[
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*YRHEQIRXEP (IRMXMSRW
)PIGXVMG
TSXIRXMEP
'ETEGMXERGI