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7 Electromagnetic Induction ( )
7.1 Faraday’s Discovery (Faraday )
1. The power which electricity of tension possesses of causing an opposite electri-cal state in its vicinity has been expressed by the general term. Induction; which, as ithas been received into scientific language, may also, with propriety, be used in the samegeneral sense to express the power which electrical currents may possess of inducing anyparticular state upon matter in their immediate neighbourhood, otherwise indifferent. Itis with this meaning that I purpose using it in the present paper.
2. Certain effects of the induction of electrical currents have already been recognizedand described: as those of magnetization; Ampere’s experiments of bringing a copper discnear to a flat spiral; his repetition with electromagnets of Arago’s extraordinary exper-iments, and perhaps a few others.Still it appeared unlikely that these could be all theeffects which induction by currents could produce; especially as, upon dispensing withiron, almost the whole of them disappear, whilst yet as infinity of bodies, exhibiting defi-nite phenomena of induction with electricity of tension, still remain to be acted upon bythe unduction of electricity in motion.
3. Further: Whether Ampere’s beautiful theory were adopted, or any other, or what-ever reservation were mentally made, still it appeared very extraordinary, that as everyelectric current was accompanied by a corresponding intensity of magnetic action at rughtangles to the current, good conductors of electricity, when placed within the sphere ofthis action, should not have any current induced through them, or some sensible effectproduced equivalent in force to such a current.
4. These considerations, with their consequence, the hope of obtaining electricity fromordinary magnetism, have stimulated me at various times to investigate experimentallythe inductive effect of electric currents. I lately arrived at positive results; and not onlyhad my hopes fulfilled, but obtained a key which appeared to me to open out a full ex-planation of Arago’s magnetic phenomena, and also to discover a new state, which mayprobably have great influence in some of the most important effects of electric currents.
5. These results I purpose describing, not as they were obtained, but such a manneras to give the most concise view of the whole.
Michael Faraday "!$#%&(' ()"*+ 1831, Faraday --./0132"465378!9#:9;< ”Experimental Researches in Electric-ity”(1839, London) =?>@AB78&' (132+C%"# 12 DE<FG<(HI-<8&' 9J >KF-G< !# Fararday +63L-MBN&LO3PQRSTUV<WXY 0Z>/03678&'
Faraday + ”Electricity of Tension” -[\< ] . /: 1. 7 InductionX
3 7^_/-0\3a`bcN& ]ed3f 3gaE7+-\ h3iaj3ZAk@l 3mnJ & ] %oqpr 6st!#%<&' Farday +uv9/6#3w3 df gZxw X Akyl mz % ] 6| X( !0-'
1
w< &y Q+ Oersted :T m9J 0'"s J 0w + Volta 78 : Y N%wN + Galvanometer( 3w ) 78<!0' + 2 3"! #%$&-('*)+<< 4*,7"-.c&n /</00+ Y /21+3%5476987;: meJ 0//"0+=?> 7Zk@#a%&' ] k+ Y 4"R(+ X <<(+ /!y==?>@; "A#B!"#C$NDE Fa%#GaL HI X0JLKNM c "s J 0(FIGURE 7.1a) '
Faraday -<FG<O*P9QSR X FIGURE 7.1(b) =?> (e) T%0'CUV+;- XW;X &NY(Z00[\ m%] :%^_ XC` X & ] k0a;b0c m :(d/#ef m XCg X &60X 6F-Ghjikl ]z !#%<& Faraday mon &@< XqpBrs k78&'
2
< X v K Faraday +aw+3-<O3P Y Q mz % ] X !0-' z !"#C$ X Mt!#-FG /-0-' FIGURE 7.1a +d4a7<8&: 2 g X k:&- ] g != X" J #-g$#%&' f( 1N$/#:) '(-g-w X* CN/0- X M !K0',+"'(-g+( X z- :K_a%"w XwZ/0A J u Y w"+<+. J<z = !0'/(=</:68&@FG X,/ !n#a%& ] k:K"w<(w J X +-3=3c=0. J 0 ] L ( %0-'"0:w X !?!0Y Y +3'. J &"-6L ( %30' J X12 /0>-P:%"w37a+ z 143&T-5/ac<&K6w--gw J & ] ke-768 ;F k-& ] X Y z 1:9';c & ] a7 k0' 7 Faraday <A&yFG=>-64 / meJ #a% z %0?<q+ X ) mKz !# $-x " J 0 Y "7wj X jtkA@B0 ] 78<!0 (FIGURE 7.1b) ';<+3wj+C%D$ O"w<-+E mz % ] 3= !067 Z zFG X /0"78!0-'"w-w J X 0 ] k:n- meJ 00D$ O"w!n#?<0+-- /</:K"w X ! & ] k-+6 m9J &(w<N-!#*+3HJIk/c& ] X /0-'
Faraday N&,KF-G H'L X NZq'
In the preceding experiments the wires were placed near to each other, and the contact
3
of the inducing one with the battery made when the inductive effect was required; butas the particular action might be supposed to be exerted only at the moments of makingand breaking contact, the induction was produced in another way. Several feet of cupperwire were stretched in wide zigzag forms, representing the letter W, on one surface ofbroad board; a second wire was stretched in precisely similar forms on a second board, sothat when brought near the first, the wires should everywhere touch, except tha a sheet ofthick paper was interposed. One of these wires was connected with yhe galvanometer, andthe other with a voltaic battery. The first wire was moved towards the second, and as itapproached, the needle was deflected. Being the removed, the needle was deflected in theopposite direction. By first making the wiresapproach and then recede, simultaneouslywith the bibration of the needle, the latter soonbecame very extensive; but whe the wiresceased to move from or towards each other, the galvanometer needle soon come to itsusual position.
As the wires approximated, the induce current was in the contrary direction to theinducing current. As the wires receded, the induced current was in the same direction asthe inducing current. When the wires remained stationary, there was no induced current.
a7+ Faraday 9J >(F-G7 m !0V-MB X ^_3c&'6 = >,& ] + X %#a%&(\ 1 -=?>K>a78&' Y K !] /6# ] :?!#%&=v>e3N! X ?k7"c ] 7 k& ] B&' #"%$/0?!# W r ] :"0/# Y'&!( T)<*+a= >@+, J #</ <: ( < A"BN/06*N- e Faraday m%naI s #% &!. X#/ (& ] ) “to give the mostconcise view of the whole” 05 X Y'1%2 %3 X &4 ,30 ] BZZq'
7.2 A Conducting Rod Moving through A Uniform Magnetic Field ( 4"U5 X 1 g%6 )
FIGURE 7.2a +78 z -g6- d#9@a#a%&q'I ] +:8;3c&Ik4(< v
7 %#%<& ] := X> /#%<&' 63 1@?&3+3&3Ta4= ?&3Ta4U z AB a`b /"#%&"' z +BCEDFG0# X F F @H k zJIKL #EMn7-Mt& ] 7 k&"'NOx, yz P F
X I%K%L #QM([QRP ] c &'g%6 z A J C:SP F 7+(<+c 10T4U z A%B B 8&""78&'
g6+U %0 ] k 1 \aV%W%XZ J &' FIGURE 7.2b 6\ q #V%WZ$6 ]ZY = m9J #%<&u[ z \UVW37 Y A%B B 3x X 1$67%
f = qv ×B (7.1)X0\ &'
4
FIGURE 7.2 > mnJ 0 B]
v *' I7+ q z >' + x ' I?k: 1$-g x7F Y 1 "\378&/#a%&(SW+< x 'I 1 ' a\ A&@=:a"\ A&@=:8&%+ ] YZ A&@=+ z 1 >1+#4 ]yz & ] 3= &'
g634 < m 7 %#%&C z !#%& ] k+ (7.1) 7<S f +60a#7 eJv] 1 I ! #"%$'&)(*,+-./ +10+234!6578+9:;<=?> !@BACD0 CE6FG517IHG+J2J3@BAG:KL!L517M NPO *.RQ : 9 f :S @PT?U = !VJWX!Y (FIGURE 7.2a Z b [,:J\!]6!W Z^C1_ NP` Y ) Ma ZXb?7cVW,!Y ( \J!dJeG!W Z^C1_ Nf Y ) Mg@BT
5
U C Q3L4 7R_ N . gS @PTGM5Q@BA E HJ;< = ! > !G4[J[Rb
qE = −f (7.2)
" +6Q [@PTL!6:3! #" +B@BT$&%X: = !')(+*-[:+ Z = !-,&(6Mb%@A?U). Q3/,&([X!@AX:.0g,@T " SX@TH1.#Q@A "32 N 2LQ3@T6: = !54JYM&6!7M589'&:?Q *6-P[X:+;IZ = M=<#_ N $%#:?QB!B[!>L2: (GQ35,(1!B@A!?A@H FIGURE 7.3a [6(6Q3FIGURE 7.3b : = !g @PTHA#_ N 2GQW,!Y!-BDCX\[G( &cZ;J<EFGL!B@PT$% " ;< > , !B@!H 9!I?!?A@GUJ9K N 26Q34%.HDL F [6!(GQM+NP[6!OP [6(Q3
6
4!5L?U ;< = "6M L F’ C 0 N 1517 3 = UK/ 00+:QG" L F’ [X:FIGURE 7.2c MJ K 65?78M B′(v H 21": B " C :G2 ) 5Je!B@JA!L!! G+A!LM#"%$ CD0& 0.RQV O6+P@BA
E′ = −v′ ×B′ = v ×B′ (7.3)
H' (+:Q34!L,M;< = U()$6Q,":JZ+*,#K N 2XQ%;< = UV=OX+B@A6!'BM- "2'74 "[,(,Q3).!0/ 1GZ%;< = ! > (L!@A?U3254M :6Q5?7 +B@BTG!06 7-H;< = EFJ[E8 Q3 B′ :#*9,#K N 2GQ@PTGM:#:L!;< b3?5>= +J2J3
7
FIGURE 7.4a :&L F’ [6!P@!H 9I U-J+:,\ " Z = !Y?!B CX\P[6(6Q3 FIGURE 7.4b[ :;J< = !1'(,[G!@BAH2 4[G(XQ4,"UJ0K N 2GQ3
Lorentz
(v/c ! 2 KG!1J[,(,Q3 ) U "L F [X!(,Q,M-GQ @T$&%UJ : FIGURE 7.3b : L F’ [ 0.RQ!1"%V [G(GQ3@BAL!-O1PX: F " F’ [ :6+6Q3:G+ 0 FIGURE 7.3 !@BAG:=EF @PT6!-.RQ @BA+*-B[X(XQ H6ZBVJW FIGURE 7.4 !@BA6:5EF @PT$%!-.RQ @BA'" L F’ [ ' ( :Q V O+P@BA![G(XQ CD0 [G(XQ3
L F [G!X: ”= ! > ( [,:@BA E = v ×B H O *. N KD_34 .H,@PT q U= M5<#_ N CD:9 qv ×B "7 P$ & ( 7c9 qE = −qv ×B U@T6MC1 N 2
Q3 ” " 7I3
L F’ [G! G: ”= ! > (X[X:@PAL:+2J3 JZL44MJ:V&OL+#H(GQ-
.0ZB@PTJH62 N 2J+J2![J9,:)' (0KB+23 ” " 7I3X0 !!#"?b g0KB2J3
8
7.3 A Loop Moving through A Nonuniform Magnetic Field ( V OJA > U )
FIGURE 7.5 [&J K 65?7 M;I[X2GU). & V=O B> UV L!
[&C K&1"B :H ?4Q*#7 C,3
4!M9$Q=JM: L F’ U K N 4!LUV&OG+@BA!'PM#-G210- 7 +Q CLU3"$LQ-*J-[2&*R7I3"10 C,MW G!!:6Q 2 "GMB@TH O *P. Q-*-[(GQ3
*H1L F [X! B H N " M:#AK+J2H%$DN " M&'( d [X(XQX" "%$15?7 34 .1UE :DJM FIGURE 7.6 [,:)62*,+- .0/ [&. _ B U1G2 N 4 7 3
9
,@1U+:Q @6M+H. *,+,- . / : L F XM . N 2GQ3 ( L F M:P@AHX+2-K=HGZMA?U/. Q5 M,Q%*+- ./U 7 Z @46M:@BAJH' (+:LQ3DKC KBZ64@BAG: !6M:#"%$X+&[D:Q36Q2J: *%+-.0/&@17< U@TH%2 4[#6Q 4'U(,CD)&*+,'M96Q4b[.-Q3 )
L F y W,^M v [ 4/ H&X2 N 2,Q0%-Z Q t M2 N W f"G[#5C.-1%H B1 Z ` "G[15C.-%H B2 [6_KX517 3 &G6M&GQ@PT q M 9U f E0KZ f &7&2GM*GQ/I43$1U56+:Q :7 WG^PM%8!6+ 2 "G[,: f :9!I: ds <;=K N 2GQ C 0>31$?@&A :)2 4 [GQ3B 2 "CD0@&A :Z%C"1 %H w *BCD0
∮f · ds = qv(B1 −B2)w (7.4)
['GQ3
bK) 1N[ q H6UV2 /D0). N ZL4) 1N > M:?4'D-JH * 0 +J-./!G-I431$G: f M5_ N + . EF1UG$LQ3HI&JK'LMN EF6:
1
q
∮f · ds
[XQ34 . U electromotove force( O J&P ) P5/Q3R0S E U ,2Z@ K N emf 2 7 3
10
SI L[DG: V [XQ3E =
1
q
∮f · ds (7.5)
electromotive force 2 7 X:eGM 4 10 [[ N -3.,- : Volta @1U /[@BTU/JC:&PM D@T!>/EGFK N 0K P3J44[ emf UBAK N @T U6QILM=<#_ N 20) Q 0 Q 9 [ - Q,4' M+K3 H& R U6b% R0 Z emf E : Ohm ! LM+K!H#_ N @ U: :I = E/R 3 "9$ N 26Q "[,: f :A > U62 N 2GQ@PTGM 9J['# Z E :
E = vw(B1 −B2) (7.6)
6QC.-1UGb3
4 emf :LU%$ Q'&!(!)+*,-./+0 21+ 4"%$GM&XQ36U%$RQ> :ZL4LU%3 :LQ)FG[ B F31$1U45!:Q3
FIGURE 7.7a 6I1XQ2: C U%$ Q> Φ : S1 G[ B F31$[$0Q :
ΦS1=
∫S1
B · da1 (7.7)
C U%3 :Q/FX:#76M&XQ3 FIGURE 7.7b M:) 1F S2 H,1CP. N 2GQ3
11
U/56 :LQM:XMFU K N b 23 ∫B · da :?02?QFXMAKV
1UG$Q CD0 J34%XM2 N U0"9$ N 5 7I3
S2 U$ Q ,:∫
S2
B · da2 [',Q3 vector da2 : S2 FPCD0 G? - N 2GQ5178MB_640 M4DR5 3%.D7 :./ S1 F#! W "FBM 6Q3 C U%$ Q g5HG?^'-M 6Q' - G:g1! ' 6Q3
ΦS2=
∫S2
B · da2 (7.8)
6 1 2 [1!#%$6:)2 4[XQ40U'&(D* :
divB = 0
Gauss CD0 S H%)41+!-F ( *,+ ) -/. V 0.1243 4365∮
S
B · da =
∫V
divBdv = 0 (7.9)
75'89;: 0 S1 S2 0=<?>@A,- -5 FIGURE 7.7c BDCE?FHGHI,J6KML !!N7O'PQ LR8S2 S -TJUWVHXZY\[T] C 0^_54`\ab0cd/365J,O'P>e vector da2 f ]g,h6V/X-Ti5'8'j7O
0 =
∮S
B · da =
∮S1
B · da1 +
∮S2
B · (−da2)
k7lm ∫S1
B · da1 =
∫S2
B · da2 (7.10)
-/iT5'89Z: ] C 0?^54`ab0c d/365"nH.oJ'Nb0"P Q L fqp\rTs%t 9 f 0?u7vHw'A t 5'8
12
w div B = 0 %`%aTOHwAM57J b0uTv3W5 9 f - X : J 9 f ] < r 5'8o s -
`\a/]e f"!#%$ @%&_>?A('_5'8
`%a) (FIGURE 7.8) ]'`*[,+-,N S O%JH> A t 5.TJ"/07J 1 54N2/iH58 9 J!N0?^ m43 @/5 `,aH]%57682/iH5!89;: Lqo<%=7J>I?@<%=[7J!aW08A,2 t 5\KbLO'`\a)H]%BWJ!`\aM08A;C J fEDF 5 9 f
2_X5'8G%H J"`%a)60JIM>?e m4K mML CNO?[b0JP < f o s NO[2 ^ 5 `%a] !# TO s
5'8
13
<TO k t A L f div E = ρ/ε0 2/i/5J2<5 6J Oh6w'AHJ(< E OHw'A !# 9 f F 5'8`\a/]Ho 9 2 t k 2
5 6"2Ti/5'8t A t 5 t @MIO m @bI60?^5`\a7JM0WKMLR8 dt $ @ 375 f @6I%] vdt $ @
< 8 9 J 9 f O6K m @6I/0^ 5 `,a/O k tA 2 !#"2$% Q 9 5"8#& `a] @7I'J(75ZN S 2J"N)*
∫B ·da 2i5"8 FIGURE
7.9 ,+r 5\KWL O!`%a]-.7J0/*2 B2wvdt $ @'1 F 5 2.6J0/*2 B1wvdt $@3546WK%>?A87'J dt 2TJ @bI970^:4`\a7J$;% dΦ ]
dΦ = −(B1 −B2)wvdt (7.11)
f?s 46 emf J<>= f 7'J`\a7J$;%TJ<=b00?>@84 f A7'J'0WJ emf ]
E = −dΦ
dt(7.12)
f?s 4;7 f r 46
GB OHo sC J @ I9/O/wA 7J9D X0E/O9D s <GF>HJIK47 f 0'LI87
f 2 X46FIGURE 7.10 J @MI C 9M t 2 C1 JNOTOi m
9Mt+ dt O,] C2 JNOTO
t
A t 4 J f IA46 @bI S J%i44[P ds ]Q> v 2SR,J'TMw t NOTOU>e
J f wKMLV6
14
9Mt 2(@bI/J9(K4?Nb0 S f < I6 7'J
9M 2/J @bI70^544`\aT]
Φ(t) =
∫S
B · da (7.13)
2/iJ46772TJ 0D>?A t 42]Tw'A t 4 sources + `\a B 1 + : A t A 7J2TJ%uWJ.2 B ] s $;%H]/w s%t 6M
t+ dt 2] 7J @bIHJ9( 4N/](,O : eJ\JN S O dS $ @1+ : A t 4 ( @bI70?^:44`\ab0!cdIA4J,O @bI S O( + : e7JNb0P L 7 f 2 X'eA7 f 0 t 6 ) 6WK%>?A
Φ(t+ dt) =
∫S+dS
B · da = Φ(t) +
∫dS
B · da (7.14)
j7O> dt 2TJ!`\a6J'$;%/] 9 :\L;o dS 0?^544`\a ∫dS
B · da 2Ti46 S 2TJ! ,N)"P da ] (vdt)× ds f <! 4J2N dS S J'N)*/]NO"[ C S J![)>* f w'A$#7@S4 :
dΦ =
∫dS
B · da =
∮C
B · [(vdt)× ds] (7.15)
dt ])>*TOHw'A]% $( + K m $ w'A6w l L fdΦ
dt=
∮C
B · (v × ds) (7.16)
15
o s 3k J vectors 2 3 ) a · (b × c) = −(b × a) · c 0eIHJ%2A7J=T]#DX A
dΦ
dt= −
∮C
(v ×B) · ds (7.17)
0:46@bIOD>A : 4 < q O < *] q(v ×B) 2Ti46N<6i%e m JS7'J*08@bIOD>?A'[)>*bw'A + : 4 emf ]
E =
∮C
(v ×B) · ds (7.18)
2/iJ46(7.17) = f (7.18) =W0 ?>@K4 f TOTe (7.12) =W0 4 S ,u6J C Q K @OhWw (7.12) =FHIA4 ( 772 On7wJ@ & ., !# v 2/i4f ]"! Mw s%$# eS7 f O&%%u$' ) 6l f 4 f f/q s 4(N<Tie m *70 w t 4 @ S 2 [)*6we emf]K7 @60?^544`\a $%) &*,+,- O.bw t 6
[)* VX f `a / EV f ]-021 234,5 +76 t 40698 F FIGURE 7.62H]%@:9W0"^54?`aT] / 2/]iJ4J>3<; w t>= 6 (7.12) =7O? r6 4 - + - 0 DF O& 6 4 f -01% + @@9 S / < 0!g c m O I@'BAqO emf C # t 47 f r 4 (FIGURE 7.11) 6
7 &+- f EHV DW0DJ4 O( t EY7i46 FIGURE 7.11 OL 6 e'A O%<E F G <,*OH' # E 6 4 f X7 < EJILK `,a $%60M<NK4 E/V @0^544`\ab0815467 7 f ] + - f E6V,OIK4OP;%2/] sQ= R;\O\] HS s 7 f + 26i46 $%/O * + AUTbV ? r6 2Ti46 7 ,V&W 0X<Y \O Lenz( Z ["\ )
Y f
16
t AV6
Lenz Y A G k 8, FIGURE 7.12 OP 6 t 46
@WO:'4?`;0 6w t 46 0"^ 4`aT] 7VTX82 f\f 16w& t = 6 7 ':A s `\a $;%W0M,N 4 ] S V/X `\aT6e " s 467 `\aM08154 ':A ETV < E E 6 4 ':A s 46 Lenz
YF G <*]K7@A t A ETV < E E 6 4 ':A C # t 4;7 f 0 F = 6 46emf FIGURE 7.6 7.11
L 6 eH'A s < E 0@ 1:4 f XJ@ ' m [ 2 08@! 8#"$ 6 4'6J2]%,57 ,08@! %0&#')(+*bw t 4$-, A 67 #./ 10 F 4e < E FIGURE 7.11
2@70 E 6 4 f XJ7 < E = *60 D F '@AG62 E C -#3 ) S 2 `%aH] B2 2/i # 7
]-WVHX *6045 46 23 ) ] B1 i m 2WVHX 6 I(*604)5846 B1
E B2 ' m MX t 2@ = / v *]2WVHX f4s m EHV f ]789 2/i 46 ' # K7 @T0 G
Q2 84 ]U>/ +;:< 0\w s 5 6 s + 5>A7BA w"+= #K+ 6 e# 0 @!">)? ] [@ C>I 4BA fs 4 2Ti46 w Lenz
YF m H6e s>= s m \i4 t ]@: = *,@ = ETV = s8+ o A s# 7w l A #C)D w'06
<FEEG<+H)IJ =LK%r6 4BMJ f w`N2 POMw t 4 @Hi4 t ] @Wi46 FIGURE 7.13 7 6 l 2#QR7w('WeS7 f 0 K$# ':AV6S%]1T7o
17
G s `;0 G Q 82 1O w" t 4 G @@60'L w t 461EG7 s ([ O.S %P 6 6
6 2 @ B # S46! #"$Q% ω[rad/s] &' # ()+*'A-,/.10 2#3 t & ! #" 5416 $ θ &718:9<; (>= ? θ = ωt+ α & @( ( AAB& α t = 0 & C !ED "C 46 /FHG$%I& @( ) ,!JD "K LM1NIGO( B CPQ B sin θ &@ (,ORSUT 2V3 t & !JD "W X>(BYZ
Φ(t) = SB sin(ωt+ α) (7.19)
&@ (,OAAB& S !JD "C5K1[& @(,A5CE)\^]_`Ha1b
E = −dΦ
dt= −SBω cos(ωt+ α) (7.20)
)UW(,!OD " FIGURE 7.13 CWREcdL5efhg^ijV&k1l1_mn)o ; TI ;Ip1q1r1s t NuavJ/w 8 GO(IA)xI&y\IREcz,A+C 6 :c|h~H?1O@ S<TIROc-, FIGURE 7.13 &HC !OD "C+K[1 80
cm2 ?UYZ1% B 0.005 T & !OD "HC'51%1 30 'I&@H()^G(,ACJ)V\5?ω = 2π × 30 > 188 rad/s & @(,emf C+> !JD " L]_ 9<; (B`Ha1b C5I
E0 = SBω = (80× 10−4 m2)(5× 10−3 T)(188 s−1) = 7.52× 10−3 V (7.21)
)UW(,
18
7.4 A Stationary Loop with The Field Source Moving( ~ LOR(WL *T ( !ED " )
FIGURE 7.6 C E !ED " ) L#TW( F’ &GJ(IAH)x&>\1(,nc<*T !>("!$#&H % ~z? WL& ('(1 !>( )#& @(, F’ C*+W x′, y′, z′ & @(BW? !JD "I, *T (,
F &-W*T./10 ! )^a2 #% v′ = −v & −y′ 354 LTW(,2#3t′ &C !JD "HC 2 6IC87I&wyS8./YZ11%O B′
1 ) B′
2 )^GW(,9 F’ &OA5C8:7C; < La =>GO(,E′
1 = −v′ ×B′
1 = v ×B′
1
E′
2 = −v′ ×B′
2 = v ×B′
2
(7.22)
)"%OSUTW(, F’ C?w1@OL )SUT A ; -ACB1CaH&@ ? a&%1I,6J .I0EC DE mWLF c E′ CG'm1[QHJIVL1 %y=% , K1LyAC M 3NO D,PQ R GIGO(om1[
Q ∮E′ · ds′ = wv(B′1 −B′2) (7.23)
)"%W(,(11) SWCG'm[QJACG'T U < C emfE ′ )RWVE(, *9XaYZ[ AC8T U QR GGO(%>= E ′ OACXaY1Z L% 9^; (\OC ] 4 a-X@ .nxC ^I& @ (, E ′ JACO DPJQ X (UYZOC_` )abE*#T$(#,OA#C ab Q-c (#L? O DPedf $*
T1(W[g hi jk l C+% −v &5T1(#C&@H(A)oL m10HG ;p R9, 2
19
dt′ & O DlP C : 7WL1# (BYIZWC[ dΦ′ = (B′1 − B′2)w(−v)dt
′ L%OS TW( =E ′ = −
dΦ′
dt′(7.24)
) &y\1(, & B C $ [ *T ( F ) O DlP [ *T ( F’ C 2 6HC8 Q
)ORycz, F &C ?w @
” t L1 R &H %[ 2 t LE*% YZ% [@(,H* *#a %1,_1mWC O DP [ A+C+Y Q 5T1T O DP < C#aXL] 4 aXW@. v×B [N] Cb-[1|T (,A ; C O D,P L,!.("G'm[Q[ emf E & @WSBTH? E −dΦ/dtL*l,C 2 &wES. 23 t &C O DP [ (oK S
< C+K[Q ∫B · da [YZ Φ &
@ (, ”)" c-, F’ & C ?w @
” _mWC O DP [1T1(+,,1*+T1a,) # [IalX Q GV,WA B#La, E′ [ @H(+,"!2 L % $#YZ% B′Q&% (oY' t [% −v &TTa [ % = ; T
(IREc L(! ; (,A+CValL *5T( O DP5Q *)1( ∮
E′ · s′ lHJI &% ~z?IA5C O DP5Q X>(^YZWC 1_-`OC − +WL"J* ~ %OSBTW(,,C 2 &w S . 2V3 t7 & C O D,P [-O(UK <&w 8 &y\1( B′ C+LOR1SUTYZ Φ′ ∫
B′ · da′ %W(UK1[QI&@ (, ”)" c-,A5C. OA ; &CJ)A/10 2 t L q3 4 @65,7 B 8 B′ ? t 8 t′ WC,9 Q m:<;
T>=-? p@ :OC5 9 v ≤ c L61TPA G5"7 ; v ¿ c %>= v2/c2 BC 4 \ T B′ K < B L<;n~ %D5;? t 8 t′ C,9 ,.,BC 4 \57
7.5 Universal Law of Induction ( EF t %aY]_OCHG I )
FIGURE 7.15 L"J<;-.,KL Q = SUT 3 6CKM Q&N cO7 PH=UCQ SRT LCHS<T1VTU] L+ GOA8 [ 4 \>5.71VWHL X ; .>YZ O DP L[\OC>]^u [ef 9o; T$_5"7"` ; . $ 4 aVYb; 4 \%1C4 ?K1LHL1 S-8dc][ef ; ; %7gh
I /0 O LaVu Q u<;-.Q 1 9ji ?&k 4 \+L 9 v4 Q 2
Q G7A5C<8V\lmn i,o6pq>rbs7"tu 4wvdx Q Qy<;.>7gh
II Q 2Q 1<z i .H/ 0 O La#u Q u;.Q 1
Q |z v4 4 \/L5 G7 v
20
8\ lmn i,o6pq>rbs7Q z [w8\5lKM I 8 KM II
Q Lorentz R 8,;T ;.7v 2 6 KM 4 ^u [ x 5[ vdx O P Q X|5 B LOR>5BYZ 1_` 8"a
bHG5 v 8B! SUTD57gh
III 2 6 Q,[*;5T*5W[HGV e KQ G-V i T /0 O aVu I
Q ?yREc 7 v Ryc L-;T O P K Q X|5 B
YIZ [KM I 8 II4 ` 8S
; ~ G5 v 8 [ 4 \57 lmn i,o6p>qrbs67Q 2< O P )1# Q,c TD5W?w @ [ v ; YZ\ Q 8"!L a 8.;
Tw SUT*5*8\5lKM I 8 II 8 IIIQ#$ 4 \ %7 v 8\ 2 6 Q Q %'& 4 \(:S
. 8")*<; REcO7 II 8 III YZ\+Ia , L L1 -|z8%"9[/.nc15W[ R'0 ?w
@[12 t % BQ wyS^T1*58) # 4 %3& 4 c 434 [6 v S^T1*5 ! W% 4
KM Q&N SUTD5 65_7 x %78 L ;^u [ KM III4 !9 RJc L5 x;:'< r p .65>=? 4 YZ\6JRA@al
[B ~DC x . FEHG I=HJ5 v 8KG%D57AELG 12;L6FEl[!9 2 6 MNO \;P R 3 4 P ∮
E · ds 6= 0 %*5RQ S QKTVU [DXWY4 P TZ\[] v 8KG [_^K` ;Fa U 7b M III P I 8 II 8KGdc 4He s7"^fFg PHh9 ij vvj4lknmx 57
Faraday Nl^Roprq b's P v Rtu L v bwyx_z o |~ k 6;;^o'yA
emf P B MN 6F ij W k PH []6 L [ z `
Faraday '~ ¡ wR¢ U 6£¤ F¥ §¦ 7¨6
21
C¦
x− y− z z `F]; S P Cw B(x, y, z)
¦ t\
(x, y, z)' l MNO \; J d
E =
∮C
E · ds = −d
dt
∫S
B · da = −dΦ
dt(7.25)
vector
H;curl
w U (7.25)
P [ Faraday Q M! "$#% w& ¨'(6 kdz d
∮C
E · ds = −d
dt
∫S
B · da (7.26)
¦*)+ *, C ¥A w -
S !. ]3`!/ z ][ y01 2
curlE = −dB
dt(7.27)
;\ 43 *5 ¥ [~6(7.26)
6 (7.27) w " 7 C
w12; a Z $89 4(`] ] UD!:~$; Ywb<=5 >?F]
;¥6
12 7 @ B
AB2 7 []C2 w d_¥dDFE3 Cw
aG*HI/
B 7JK ¨ 'L 7M\ B ·a $N*O -P 2 Q CRS w,T]3`
curlw-UV_z o ¦ CW w,T ]\ G*H ¦ 9*X < ¨ ∮
CE · ds¦ NO YD=E 7
a · curlE\;6
^K`CDZE z `a · curlE = −
d
dt(B · a) = a · (−
dB
dt) (7.28)
w,[ 6¥A 7Z\-]^CJ E_G a .*`a kcbde5 d
curlE = −dB
dt(7.29)
f7-fj > `jf` B 7 khg!i ~ kkjl5 ¥ w-mon a` d6 dB/dt
¥Cpy∂B/∂t
&¥rq¥rq z aCs ct=fuCv " ,#%/ 2 . $wx ¦ [ _o :
∮C
E · ds = − ddt
∫S
B · da
curlE = −∂B∂t
(7.30)
vy ∫S
B · da M g 7 T·m2 ;H6F¥n w weber( z|~| )
! 1 weber = 1
7 T·m2\;6
22
curlE = −∂B/∂t f ~' w z o !.`_a7 n¨ (7¥ "¦ 7¨ q
B6 ¦ W 2; curlE
w k /FCs~Ca z q kp X; ¥dd ¦ Ew !s~ U a z q j 7f curlE = 0
f 0*1 u w! "# 5 ¥ 7 7¨6$ %'& z a FIGURE 7.13
q f)('*,+\-. 7!f 60Hz( /0 60 12*'3+ )
4 .u. w . z 5 z q (FIGURE 7.16)
u.6/7vy8 7 sin(2π ·60t) = sin 377t z a 5 u. 9: 7;!<=> H
vy 8 B@?A ¦
50 gauss = 0.005 T CB 5 q fED FF z q FIGURE7.16 xz 5 q f FG 10 cm
IHKJ!I " Al u 6,7 emfwL 5 ` ¥ IHKJ~ ; 7*`.MD B
¦)NO \; P Uz q
B = 0.005 sin 377t (7.32)
+2QR Cw :l vye7
Φ = πr2B = π × 0.12 × 0.005 sin 377t = 1.57× 10−4 sin 377t [T ·m2] (7.33)
emfwL
E = −dΦ
dt= −377× 1.57× 10−4 cos 377t = −0.059 cos 377t [V] (7.34)
E6? A 7 59 mV
\;6 SIT 7 Lenz$#%*6 UU d
ΦE6 w
FIGURE 7.17 x_z5 u 7V q f n a` CW=p q 6F curlE
WX6 : 7 E 7 6 f` z 6 zKY Z[C 7 \ ;< w] HJ~\dHJI '^ 6~_a` b u ¦ f Xd> E
23
24
7 H I n a`aHW ?7¨ 7 N U \;6¥ ¨ ∮C
E · ds = 2πrE = EE 7 D q L _a6` 5 6 cu ~?¨ 7 f Y # HKJ!I_ u 7 ;H 7 FIGURE 7.18a
q f ¥ §¦ ;6D z u _a` b ¦ > O 7s K¥ f d & > FIGURE 7.18b
xz@5 q I /\ u ¦ H q f ¨ 7 H J N u 7 ¥ 2 . u ;/ u " A 5 u _ "# *( f 6
7.6 Mutual Inductance ( " )
2 . H,J 5 f +Q!R C1
C2
¦ n5gCi U a` (FIGURE 7.19)
u ;3'?7¨6F7¨ u. I1
¦ [C1 . a` 6
C1 K. a!` uC. ¦ I1
` q N U AF ¨ _ v H vCy8 B1(x, y, z) 5 6 W!a [ C2 :l
B1
y Φ21
' W q
Φ21 =
∫S2
B1 · da2 (7.35)
¥¥XS2 7+2QR C2
- 5 6Φ21 7 I1 #" & 5 6 W @ 7 U A 5 2 . [ gi 6
Φ21
I1
= const (7.36)
I1
¦ %$ l ¨ W F ¦&(' )+*,-cfl > C2 . N VXef 2D B1 7 C1
u . I1
7 U ' u. # 0/1 q d 5 WFp q ( V q !a ¥ Xef2 E ¦34 !f 6 !7 C1
C2
¦10m 5 a`_aF 10ns D 76; C1
;u. A ¦ 2 8/2D,f q f 9:Zf ~ # 7;< ( )
v*y Φ21
' 7 I1
25
' #" & 5 6 [ C2 " Al emf 7E21 = −const
dI1
dt(7.37)
;H6;¥6const 7 (7.36)
~const
7/N \H6;¥A M21
' 5 (7.37)
7 ¢ q & X~ :E21 = −M21
dI1
dt(7.38)
¥d U @M21
Mutual Inductance( )
` q _¥dA 7+MQR / g ' 6 M g 7 H( Q )\ H = V/A = Ω·s
;\61A/s u. ¦ F 5 ¨ C2
~ *" u ¦ 1V~6> ( *M3 !7 1H
~6& x !aH G f!"# C2
?¨ f!$# C1
¦ /N&% I ;C< $(' !a` FIGURE 7.20
[ *) 6
u. I1
¦ . a` C1
[ ;< v*y'8 B1
?7¨ 7B1 =
µ0 I1
2R1
(7.39)
; 5 6R2 ¿ R1
P U !aH G `!+# C2
,.-F~B1
J*K 7¨ 5 6_¥6 ¨C2 :l v*ye7
Φ21 = (πR22)µ0 I1
2R1
=µ0πI1R
22
2R1
(7.40)
26
f 6 n a ¥K & (7.36)
F”const” 7 µ0πI1R
22/2R1 f A . C2 C" _ emf 7
E21 = −µ0πR
22
2R1
dI1
dt(7.41)
f 6U*23(( C7 µ0 = 4π × 10−7 H/m
WC6 E [V] = −
2π2R22
R1
· 10−7dI1
dt[[H · A/s] (7.42)
n a M21
H R1
R2
m!' 5 d
M21 =2π2 × 10−7R2
2
R1
(7.43)
;\6 * . X a7!` ¦ ¥¥ 7 7 AX 7!f` emf
C2
*" 5 u . T 5 ! 7 Lenz #%!#" N%$& ` S%'
C1
N ! $# ( f*),+ N1 - /.10243657895;: +=<. I1
5 3657 DC;C< > B1
(N1 8?@)BA /O C2
C 7N2
- FG R2
"#ED 3F 7895;: +9G $# emf/H 1C1I
emf A E NJ - % N2 8
emfK 3L5 "1M 7
2 N OPRQ Multiple Turns( STUWV ) A M 78 X;Y EZ[\P]=^`_a]b (M21 =
2π2 × 10−7N1N2R22
R1
(7.44)
c 1de 3L5 (7.44) f c ( + 2 Ng OPQ g - 0h(%i - D 5>7>87 - 0 g1j=k (!OPQ g%lm En;o 7 7Wpqsr/8 Ltu rv578 w D w 2 Ng1xy z g t1u|/~hd r A@) 7sp + Z;[sP1]E^`_a1] b M21
( u 1 r578 M21
( xy 1 g/;; <% I1
h"M 71 x9y 2 g emf [V] I1 g9; [A/s]
g n c NhWM21 [H] =
E21 [V]
(dI1dt) [A/s]
(7.45)
u1 c >
7.7 A Reciprocity Theorem (Z; u1
)
27
x9y C1
C2
+ C2
1 ; <= I2
5 > = +%x9y C1
. < E∞∈ z g = Z[WP]^ _a] b M12
(
M12 =E12
(dI2dt)
(7.46)
Lu1 r5 Any( )2 Ngxy N 8%7sp
M12 = M21 (7.47)
8 o K 365z g ( "! #$%;D 3 K 3L5&('*) c ( A 8 FIGURE 7.20 g,+ A.-/
A.0 c r e 2 Ngxy*1N 8%7 #2$% (.324 w A 8 M21 5 g R1 6 R2 g87 :9 ; <(7.47) fD 3 z g Z= A xy>1N 8%7 +
M21 =2π2 × 10−7N1N2R
22
R1
6 A D 3M12 =
2π2 × 10−7N1N2R21
R2p@?A BC 1 6 2| o 7EDFG ed p g1 ( IHJ K
u(7.47) f |LM dEN 1 ( + xy C2 O g<E I g 6 w 7 C1
|P QRΦ126 x/y C1 O gE<E I g 6 w 7 C2
|P SQRΦ21
T w 8 z 6 |JU r A )951: A 3 A8
Stokes g u1 1,+ 6 B g Vector Potential
A +NhW B = curlAcWV 6 F∮
C
A · ds =
∫S
B · da = ΦS (7.49)
| K That is, the line integral of the vector potential around a loop is equal to the flux of B
through the loop. ( XYJZ[.\E]^ Vector Potential ^._`abc d,e(^Z[\fghYijk.lIm K )
n6 o cp d + 1 Vector Potential qr1g%<>1h+ tsEu 6wv g>+ 1x.y w 78*
:
A21 =µ0 I
4π
∮C1
ds1
r21
(7.50)
A21 q x9y C1
| 5z <= I 1,+ M 7|65z~ (x2, y2, z2) 1 )" Vector PotentialcWV ; ds1 q xEy C1 g q cWV ; d r21 q ds1 g ~ D |@~ (x2, y2, z2) c g
28
c"V
FIGURE 7.21 q 2 Ng Q C1 6 C2
|JU w 78W
(x2, y2, z2)| Q
C2 O g V ~ 6 w + z g 6 F C1 O gE<= I 1>+ M 7. | 5z C2|JP SQR qΦ21 =
∮C2
A21 · ds2 =
∮ds2 ·A21 =
µ0 I
4π
∮C2
ds2 ·
∮C1
ds1
r21
(7.51)
c"V 1 C2 O g=<% I 1h+ M 7 |65 C1
|JP QR qΦ12 =
µ0 I
4π
∮C1
ds1 ·
∮C2
ds2
r12
(7.52)
c"V
r12
pr21
p D | r12 = r21
c"V O g gqg P c"V : GQ O 11N;N;g( q | 6 M 7 6 w! [8 g qE"`_# Q g$ 6 z g ;g 6 g n`|&%e ! Q O g C 7 1N 8%7(' w d*)`| + w 78*(7.52) f 6 (7.51) f!g,"q - z g. |&/0 order( 12 )
cWV M43 ! z g*56 | e3 p879 :; | e q w :< = 1 Φ12 = Φ21 6 : g c ! > 1 M12 = M21
'
29
z g u g2 c M12 6 M21 g | q: : + M43 z q g2 Ngxyhg Z[WP]^#_a] b M
|%e +< z gg u q w w ”Reciprocity”(
Z> V <q )u 6 zz cg 6 q x.y#g: <hxEy`g Z u q,1 p V . z;z c n 3 o 6 c o |
Problem3.27c"! ' d xyf Cjk = Ckj
|$# <&% w d(' p < ) Z ( " ) %"q* P q"+!g p ) g,->1/. ' K10 12W.#2$%!g34 |$5 w 3 <
7.8 Self-Induction ( 6$7&8 ]^ _a] b )
x9y C1 O 19 = I1
19; 6 F ! C1 6:9 |tP SQRp w 3 rg emf2; r QR g s;8u cV< 3 p ! v g Q ; g=34 ?EA :
E11 = −dΦ11
dt(7.53)
zz cΦ11 qx/y C1 > gE I1 1*+ QRi? B1 g C1
|P @QR cV - @Aq emfq1B*1,1h+ change( CD ) 6 q oppose( EF ) 6 : + 1 z 6 < Lenz g3
4 |$5 w 3 < Φ11 q I1 1 n 0 g c
E11 = −L1dI1
dt(7.54)
6:G )*uHL1 q z gxy g Self-Inductance( I/J1KL(MON&PLQ ) 6 +
L1
|R"S c F2 x9y!g0 6 w 3 FIGURE 7.22c U w d
(Problem 6.14 g ) jk T 9U g #V 8OW |&%e + < N X F gOY8[Z | = I qY8Zhg >\1] | g rg ~ c^ Fr
B =µ0NI
2πr
6 V de | SQR1i? | Y8_Z g"X F |tP SQR q z g Q R1i? | Y8Z gjEk c w d p g cWV :Φ(1 X F ) = h
∫ b
a
µ0NI
2πrdr =
µ0NIh
2πln(
b
a) (7.55)
N x(X F gxy |tP a` )WQR q z g N b ^ F < :
Φ =µ0N
2Ih
2πln(
b
a) (7.56)
= 1 ; E qE = −
dΦ
dt= −
µ0N2h
2πln(
b
a)dI
dt(7.57)
30
+ < 3 ! #V 8&Wg 6$7&8 ]^ _a] b qL =
µ0N2h
2πln(
b
a) (7.58)
c"V de | 6$7O8 ]9^#_a] b g - p H
cWV h|
mc q 6 (7.58) fhg L q
L [H] = 2× 10−7N2h ln(b
a) (7.59)
6 : 1 %ed !g ; g g.9 6$7O8 ]9^#_a] b g R1S 0 6 w 3 - 6 %e p :< w w ! z |$R"S 6 1% ; g > m | V 6 6 -W1: wr c"V ! < 3 6 > m V g 1"E 6 Z *|tP QR ^ 1: < 3 w z 6 ' & w 3 6 < 6 QRi? g ^ Fr
B q1% | g | r 6 6 1/r g*+ 1: ! B × k g ∫dr/r
|r = 0 c 6 s"! | cWV z | x# 1Eq ; gEl m | V c q:<%$ g ) 1% z 6 cV ! z g89 < 6 . / cV z 6 R"S'& w() 1: +* :< \,.- 1Eq V/| :< 1081 12 </"q r X F@34'4 q = : QR"5 xy - 76 3'89;: : < 3 <<>= 8>9@?A CB"D : q%E FG* H H <I@J 2 : : < 3 <>=LK"MN%O 8>9@?P'Q 2/SR ZUTV "WX5 emf
FZY\[ <];^N>_Z`Za ?b"c\dSeXf - <gh F hZi7< : I ghj
31
3 , F ^ [ i [ ^ [ = 8>93 F I3eZf3cd
2! N (7.54) " 5#$ 2 / &% d H' $5 H"[ ] * 2 i [ = 5 H)(* +3 R,7I- 3.0/\31 ? 23 2 345 ? 806 2 / = BZD H87 -?:9 J<; "[ 2 N>=?% 5A@ * 4053 2 /BC HDE 5 HZ[ =
7.9 A Circuit Containing Self-Inductance ( FHG I?:JLK8%9 )
FIGURE 7.23a3 % d FMG HA L
3NPOQ ! J Inductor(:RS
)?emf E0
?@ /T <gU dQ [ V : 2 &% =
NWOYX 7 - X ! / gU T X E ^ ] 38Z[U?RXHQ = 38Z>[ ? ' ! :H\Zd] Z[ R:H] N 8>9^ d>_ % P`I] I% :bad] [ :Ldc <>=FIGURE 7.23b
d H Z[defZ[
R?g%9Rh"d0d&]i = 0L'dIX 7 -jk P F B ml< : F @ J % <
; i XLF J>n>f < :Mc <S=
FIGURE 7.23bH8o ] T < :qprsgS9 j r>tRhfR@<
:L
LH Z[1?:uRi [
;Z[
RH LZ?:uRi [ = i] ^Pvc < j HI j &% drtZf i ig%9>fA@ <%=
g>9 j ghIF
dI/dtj Qw fef>c < : I N
emf LdI/dtFx j eZf1? y>z <H|d0 7 i <%=gU H' $ j
emf E0?@i T <%=gU F g%9I~0f h c | I ? gh j0Rj | I :Wc < : N j f j0A>j
emfHE0 −
LdI/dtf@ <%= j
emfd J Z[
R? g h
IF hi7<%= Q ! J
E0 − LdI
dt= RI (7.60)
j ? % X oR 8 i <:
32
A:
Bj . j g
(NWO jIj g
)H
IF
increasing( ): I N NWO j
Q ! J A? :Wc < :
LdI/dtfR@ <S=
B:
Cj . j g
(Z[ jLj f g
)H Z
[ j Q ! J B? :]
RIf@ <>=AW NO : Z[ j g j H
LdI/dt+RIf @ J N i F gU j . gE0 f @ (
rt"f i j f gU H ~ k Z[ Xi [
) dE0 = L
dI
dt+RI (7.61)
[ J N (7.60) " j [ l J ] T ] F ^ i (7.60) " "! sZd$# l&% d N A('*) F t = 0
d O ^ i i I g+1d-, FRI ]. T % "'/) F10 i3254I = 0
f@" '/) F O \ ^ i] ^/6 . 7d H ,8] j $:9 ; d$<>=@? I N gh H' $ j-AI0
d [ V3B % =Y% [ dI/dt 6= 0V ]^
(7.60) " HE0 = RI (7.62)d [ ] ! %
I = 0] ^ $9 g h A
I0 C j-DE HF 4 t = 0f L H @ J [ % j X =% fA@ i
dI/dtF 2(GHd [ ] ! % ] ^ fA@" MIN t = 0
jJ 7f H g hIH
K ] ^(7.60) "LMN 2 O H 23>f I
dI
dt=E0
L(7.63)
@i T I Lf g h jP = F JRQ w TS G ]
i ! fU ] i FIGURE 7.24ad ! \ &%
! V$V H `Za j ef F _ % L ] -W \( fA@" t = 0f
I = 0[
YXZ[\ ] ic(7.60) " j # 5^>_` c :
I =E0
R(1− e−(R/L)t) (7.64)
33
FIGURE 7.24bj gh / A
I0
d - s"d8/ l ] j g-+ j” $
”
L/R[ fA@" L
H R
Ωf ^ i _ L/R
sj
d [ % j X H = (V/A)·sf
Ω = V/AV ] ^ fA@"
ghI0
dI] '] ^ "'/) gh F 4 * d-<c _ % [ V1B % F4 * dc % LdI/dt[ O A21GHXd] ] ! % d [ ! " s [ L ,#X j # i ! f$8 7 g+ j :' ) % i _'& <I] (9Xd*) 7 g 0 ^ i i ' ) j .d+-, ! i /.0 21 g 3 54 j f 687 gh 9: sZd hZi =Y% ] [ f;U d FIGURE 7.25a
d ]\ ]i % [
LR < : g-+ C Q [>=?7 + @ g-+ ] ^ gU A 1 =?%
% c g-+ B "0 = L
dI
dt+RI (7.65)
fdCx i t1ADE g-+ @ ^ i c t = t1
fI = I0
[ @XZ[\ F 4 ^ i #G>H!IJ - fI = I0e
−(R/L)(t−t1) (7.66)f@ $ % l L/RfA@"
7.10 Energy Stored in The Magnetic Field ( K 1dL TNM i *O8PORQ 0 )
FIGURE 7.25b t = t1f
I = I0
H YXZ[\ F 4 M i i-#I = I0e
−(R/L)(t−t1)
d dC i g h !IJ c 0.ZdO8PHORQ 0 Z[Rf-ST i D .
dtj .ZdSTx i O8PORQ 0
dU
RI2dtV-U M t1
f DE g-+ j A1'
34
) @ i 7f-STx i *O8PORQ 0U =
∫∞
t1
RI2dt =
∫∞
t1
RI20e−(2R/L)(t−t1)dt (7.67)
f@1 x = 2R(t− t1)/L
j _ T f j c = d W ! :U = RI2
0
∫∞
0
e−xdx =1
2LI2
0 (7.68)
j O8PHORQ 0R K j c N>Of@ i ! gU j gh @z j d j j Xd E i : A= j O8PHOQ 0 t = 0U M
t = t1j .d
Z[>f-STx i i !6/7 j O8PORQ 0 X gU ]iLX j f@1 i ' s H ] P c( 0i\'d N O j gh 4 i _ d 7 gLdI/dt
d8[I] "g h h c i\"d E d H d! 4 .dtf H i >
dW = LIdI
dtdt = LIdI =
1
2d(I2) (7.69)
dU =
1
2LI2 (7.70)
j O8PORQ 0 g hIh"i N>O\d8uRi4 M i gh IJ ] ] ! % j OPO Q 0 _ U" j B d$#:U i j fR@
% g ]iAN> & ' 0j OPO>Q 0 g 11dL TNM i iIX j ] dC ]i % d j NMO j uAQOPO>Q 0 NMO j ^ j K 1&^dL T M i iORPO>Q 0 H c j F 4 fA@" gV ! f( g i iN& )' 0j O8PHOQ 0 CV 2/2
f@ i i g 1 j$&Ej g 1L^ j* K,+ E.-
ε0E2dv/2
V O/PHOQ 0 u W 0/21 ]i )3l N OdL T M i iOP:OQ 0 f X P c3 U U i 465 R f@1 78.9 f H K j :.; c3<>. O/PHOQ 0 j B2/2µ0
X Q = 4 j O/PHOQ 0 j 3 <.>$UAi ? c O8PORQ 0 LI 2/2 H U U j fA@" @ +BA f = ci\> N 7 C 8 D fFEGHJIKGL L
M,NFO,PRQS EUT)>$V B6W L =
µ0N2h
2πln(
b
a) (7.71)
XUY BZ\[ U U^] P _`I `6ab^c [ Kdegf$h _ a
B =µ0NI
2πr(7.72)
35
XUY ] PB2/2µ0 f + c P > FIGURE 7.26
>L O,P + 2πrhdr f f U MH c W
r = aU M
r = b X Z f c ["!#$ de %&' c() * "+ W ZB[-,. c ( /f P ,10 W [2! Q4365 L f879 X:; Z X=< B = 0
X2Y c)
1
2µ0
∫B2dv =
1
2µ0
∫(µ0NI
2πr)22πrhdr =
µ0N2hI2
4πln(
b
a) (7.73)
Z f8> ? [ (7.71) @ 2A6B c ["!8CD,1
2µ0
∫B2dv =
1
2LI2 (7.74)
XUY c Z\[ E D c F
1 C X f _HG fIJLKNM 3 degfOP ,RQS O1T VUXWYNZ ,1[ B c ["! ;1\ .1#$ de B(x, y, z),RQ O1T < !1] W IJKNM 3 U
:
U =1
2µ0
∫^`_ba`c`d B2dv (7.75)
[ egf c
36
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