Eg ֎LqCͽ lBKCH
7;
5p jDY;Aж@]`gOv @ ;qADek`E!41\3<@D
8h
;@H$At@e{`'2Р=@TLK
@<<F5"<D 0jȤ1
TI0T`t ~WBQ\@GE5A`KPC@6x@#^.7lP(f@4@)r`%@Fb4x
W"X& teM80,@L@@5D@
` D`@@ &pp=#* X j`Ai @ vb/а[$(hv7
l~"(~)$)`F` D@t@Hp/p;B@#4S
2f"<a k"Drr\% Z@"T͑ 9V[($9
B;xmAFJ;n Xjf`RGХ \ xAZHQP=$XB z#yf6 V@
7(h3xWHX !XAH2RO`LxJ"8/
8@9Hrx00H4ER@A<
@B?D_F/!2,dH*\ȰÇ#JHŋ3jȱǏ CIɓ(S\ɲb"ȢLHfI3P**TH%:pB*D=00xCDa!80:HW_ UAa>$Ʃ/^
$h`3
<8P)]MҀUHz#t1` H( RX h"L`!F$d #xAA) D`C414=`(Q+0
! 0p@ [A'48`+0p)<@@0p((B 5p#HPA,A h` *PU'x[hP ,p'`mXpI
A
@.`DHX8 !t80D/q
lo0 FE`4@@I
Fc+Y x`R(P0*@ `p蛯\p @l%G`
DTK/8B.4Bx5.
HAC46Pmt D
4JpR$gQ "ml`V, P %4AiU%@ @PtBAM4CI<'
EL
'0
y`A < m@@B
`
1ܰ@F`B
:6( B+X:A@PP.`&LP@_, `0 ``/)dC
8@^pKX^0!X`E(BbC`P#aT8pBxt\ \P8`
jЀ$!p@
[<,0
@MVP
0A H\ 7  !@&2.8¥Z41 Bͯ#,
L@"g`#`n;(P4,mvI<PJhLe@pAZ0@
rA̔`a6ȝ,(a2`$P# 0 ZPЪb D8Aue@6)UI. AF/&A&@U
4КWKDO
`9@^9Y
@ (Э``H7AZ0@
hGpa+´`V !V
@bJ: d +[Z`#dhR@n!QA
H`: 0&
R>\
,AhA
%04 A@@hK7Y@Y&2Sl=.}wcr4@B 1З@@ Z/0@s:
7@LeBܕj@( PЂ@?=sCJا7{6`` aHp +`S`Blll@0` *0 " )V(@AhlZPPԀdtZ`4@`RzYL2Ȅd*A`,p4X``)q9@:@h@ 4hTc9H@FjH"}4`6)j`RR
TE.0n8@$8v$,A,V!8 0:@Ba5)I}FAz @0ppEVA<,@!P
!Lakjp S
`>[0@m^
hҀ#:HuH*K^An6ȝ0x8`HXA$@@¥
x@j#@Eb7,`$ ;P1AMCFe00828}2h%g(W(RSE;q3q&!E "b>4 [7@:pF0tZ~PpLc}`H0_RWH'r83LI聅%rV .G502^KF &PJ+T@0%0M4SGz!`(aQA(0M? =6=@N@S0Ѐ4&U+q%H +PUHq#zPWcp8i;@OQ}e9o0QR@zPnR"7F("?$ZV&0!0!YN:/<09rE&01P0j7AHvf?W0_1~"a!p0V (<H7"}Ir$'7;?9L: 3;`9r
8P\F ^:00B& .pw'yd"@>@De@X#0+l}V:mptM>O8 $"@
+@#/#8)H I0KM (.J((0KPP 0y# lߤ@(+6:0ԡ,@PvsW$1pPdLQBK
O))E^pH4G+?`1)@P26e#%Hn%q&?3^HstPx,q;0=2`7@a82_ g, P)R'~h*ZV&(;@
@:!n,B @L dD7pV
@Ph'`ehr
@a07`,FA"H
=P*3w`<p
m` a.=e.\p[7*Rd{*D"!yWsN60R(j]di 6`/p3CQ@@6?YPOD0JP`4X09O w0%@WH0Subl2dBuLwQ372?83('<0I ÌA(O:jRI>@ 1KC='v3DjU5@K#ڪ
(?t(}9FЏ({ {F60Gp8H <[0 2EcT<}>^~
4!2,dH*\ȰÇ#JHŋ3jȱǏ CIɓ(S\Ң d(fˆFDMdB"dȀ.lu#,
:4xFPt\
M%p @8Q
\0uBdGd n#,Pxj˙=.$V3_: h*
7Y
L(rb?hcE$ Æ(:@@E
\0HŇ"`g)8"H~)ă$P
4@
p,#`tIxq<@Bm7p2!d= Al4MA%<H +@lj($y` <5@&4Ġ
$p1p
!
2P
X $CBT /@#C pA A "
+`A A0D@00t0C_`âM`фS)PaC464@DDt5Ml @A\@ G.D4`9M]BD +Bh@W
9@+XQ\@"x6u`@"P P0u (A04D@c\#(aD 4P.P%BEBRHE,#<JA0
P@LDJ5 <
$@
BDj@$UQ*'Tꬴj뭸뮼+k
$C:\p@L7P&8&L +(/[d=A@ hP%@5\` P`
6,ACD
;Bۀ *q
# :K̠
P.^D76@ lH0B@SA e:~"p`$@Cz
)3ă@'/  C4XIـ g`
3
``lĦ
!`3lB8"g4@_L #X@1/oo%M A"T}+@` X@z`LBgRhG`x8@@zD , J
`#<*'POP_
@&Ȇ P
0sX`X1a/ !mXMDqHG&2AHy$IlS@_C]6@q;`dZ0E
uWLw[2p&@@M`0dad7`iq VpiS>`ё]!.`"p@0DmA","J{#`Pq34>10`$r`/ pz U`Pe
ܽL}^"0GMX9NEMPJXL+'F0Y tT
/@D(06!tRPK$JIӖIa_)B:?*a20&P ]Cc֜/&%.`'0¤y@+y0(\cNU0'pkxK;Mh')y "(<P6IduŠG#p+@4AQq*QdՍD12=MuJk)M'
a0V;0v}(p5T`_w
.0C291 &Rq3PD1m=:tWg,PADH;\'E!C0,nQuo&6ym)PA9u*A{Uy7Pqk5EmPƶ15060&EPyҢY3L2FV`Z&?s)8P&0(P? !7@/#DxKLЯ!*)P
W@d+b!u*@2 A*_*:@>< pP`#@ haF.`V;ka`*j&lsGk1=j >{~^clO_4R$TQlލD,p 1}2v69@P&09 gf+"@A7y$f]"`D*'`
!g_
*'s!1X@<]A2Յ;\'Q?S
;RAv2
0 U9"}HE!QKu p0A,TAx0@4C!
$
@diK1\8$3X
RtqƈIIŊ2fad?qR!F#@â)+T#F\tQJ2ìb'D @
``!2Q%P!"d02Ќ"2X60E$ YEÄ9Rr@lxFPb
"x. Bf 20$6A,
bHA!n`$`$ZX l`
jA#!(V LH`:jAR 0`H0( Z@&i6s1{ #@*#=e
`
8z$0@*H3PN;T$0`QRBBp@ӀA8S\su O`a
@2ِS#fY:ME%
^OW\e@)R#mX#!h&pX&A+[pav`"S 6XD+.($2`>Xb $@>``BH0ୃt`zh6%Vɀ`@>S(@;<$[d`EԄ
`)! 1wlFZkEBaKaZkOP"7mk00@jU0/F*(PhC8pHe5V,0,ԈƁs;55 'ls# TW p.m3"mU7gTN*r#18`%
@@5sSu #t+!/PL!9"0W@[5@p(@*?]AGUnZkPP`g0Zc*101fam86"'2rR})iae(Dj0E.60,aĬT
@ݽ`҈hH#9@p%țb& ~c^ "@
pF=*&0Xl>Pf(*_7JLUV0KC< 9kh7Y@Z߈*T0:m@6a#?QN90EP0ȧ2pB3*2F7 /`.P0 3d8`N[Y4`e o&7XY>FpS>,`P<`#3t;j5QU.@$;1,ՄMD%z7u>ga0S`R?ưB(P>+8ev9$HX * p4`I$D`6dPI)1<`@"Kt`0ŎCq4(Ŋ @bU@"2 @HI.4bJx
x8Q@(Q=()dp&?.ANB`@K:H 1!G)tAx 1!E$Ġ$.,HD
($@"DS^Hb1 []уVp@$8A$4p@hx8@"t@Vh
`h A(` r aRx>H<@$a2R$[XJ,Z" @x`!a
Tp"8n*A,]2"R@
Ah8`@
6
ЀCc5BgV\suW^{W`v\="H (ɀ
28 M"8z+"Hdp`,@
P@@V*86sV wI,ȱ, pZp\@$DZn`d0"^\(
v<"8TTK @`
. PP >,`엄@@ 6
@hH*P8i`kT @I p@PND<@+%RBp~p(pBZ .
#T@Ml0
6`vl 08(:Yd0"41B5
t
>p
[:A T0I8 DPT "`A w$DIeIPV!
ȀcU%(X@E1hRp@+2Ny@C60 !@x`Hb`
i FH(ǒB~R@) t$@2
.ـ$pA
R
!pZp!`
$ BhB? 386R q`Ã 90;
X:AL! B A 8
TA
HA~C2.˄'4Hz!Ԁ3OB@
zY6X .@J(PB
1Vz` X@{d~0D$XN
2@2'A `
HқHGI%%F
4p
bQcp
.Wh`O
^!H
S4@93sEؠZg(Γ AHZ0+ S9(@@pI1/8LI)e(u@10E1G3S' Хg]Lf`f gH$oFO@SARr&pA#;`.@1?)`6Z =>p2 N0R##J
 0=i"}HDJlB0
~P
+$0B@c! :E #1PN֑`i()0p=!V$#`#` 0)S$5^p1 8JP=KDc @L0Q$'`1i'ofܣ)=响1)8S#H88dHp:p*` U=S!p#@!@p}esm=#36mC7@B`RN94&0&@P p"i:!@G~p1(+O;>+1`;9\'yGqft"UR"vHG#p^ &..mh<h`=@2`;
?RTP{
P+1+.9@~(P*2Y56h5,7p0E+607`ho+CdE;0!9k=h(QC=5UHӡj2ppl%Y ]r?d34U\B`qd;22^VV@2h7S5GI0jG4MA0HuIP_@3Pb;\PLP!2@g(Z{cc^qqK$ R@=Uw4@?z2,*P@r H"T 2BZeI\<@xs6z4ia/.P/BsX)UpJ9 A,p* ]~r$J>ú\*'0eY'g>\%iR+9D@6@utd&a&
CZC0R6']s"gc2{ Iprrpd5\s'5
%9W',p>!TQ:P'w>,P g"MK0bH`4p4N
`ErC)YY Vj`8HN2
2.GOPA[p/5@Qy zJ7ZUE50֕B%QU@Fb!%0'NEh*[ͣpB[A1PXCr~̋P!p&p X1ad SNuA:
KWxv/["9OL Z9;.(i 335DD#BT=G4/33s 3h)vz&8]qQS0f$7mbjrUqQB#t'2VBp@c+NP%q<7pveZI06>B}F
F 9 <_"IH_ U
п`NsŢuP67=U* .V18&[\!opѣ$@r
IV,Ty,`tpc7b;UQA0/8RKV4BCFg@EpOp160iUN/7@QmfLy,x#L'']c4x7$[3cb[R'T"zus#Cü Ђt:@ε,0^< )KRm B$p*#J.@&!o\pQ92 F)3`0
J"&@
@ь&U6k)8ڰM;9;C8B@DPDѷ@h34 LB 208}V>* f0%VE"cě\:'hr&aB_"] 'o6+K&2AP m;0`)F8X; .LC3F I? P40A`lj3 $P"JL=QP593gUQ'y:0&#P&xW'wPH%% ah(!P8kf3BࠌQ#M/bI`R*Q
z9%8@1b( 8c
8P;tAd d@qAPt0xP4%r
A \Ѐ@t@P^tP8$n5Y.m@7 P`,8P@Tj9Ah3`~@!@Pp zEF)Z`rD&߾,UH
EHꄒ!*+0 rS@p'HEh@7BBdp60dPh,P 4Evl3oeh B)nApdWoAB`4J'XkMH& O" @PFTD`pk}O9<$QjpV4m#ϡEtAOdrjXQmI憜%b
1>8ToP@D@ppx 8&$'8x%'@HPX
8P!3`*Z4
s5 6
!Ȁ2 ,(hhV#Q18'(!878pxo)mX+ @
50oH$ ob# hI hRa5Hdhx،`/0(̝@10@[FflFgFhFiFjFkFlFmFnFoFpGqGr!2,dH*\ȰÇ#JHŋ3jȱǏ CIɓ(S\ɲO 81E0qq+d!
6\h`1rXC
'L&0B8p`C5Lؠ
\AET88h 9s谙Ssom{CݷA +ؕc_Yڬf@(ҵ\,1;1 M:y3@:DQ?6@#(M@N\qv=)n ċ 1dA?ckK\;)ґ4n6)o2QֵR# 3`'5 +ED0mVT; u#,7S0Z8E?P6b"Bu)aNJPLuPI;8cmh@t`m09?a&!! @Q6(&32@,c7&0E0
&±(@fPF%/r ;sq84P`0.p0)p2403(>Sp7p$@$)3p9`Np[%[\kCV$6mLgq4>:Q&I\G(V6"p<qϡ
L/!`"Q9q'*0`! *CC"k5:Z.X'`;30(F*$:yy%p4";G[`7wnI&%el"1P> 6'=:_h87p=JxHO7ۼj+.@fX}7&p71(:UE"KzKRr=@@615[r$+$pWQր.eVɵ
Ddx?S*"
H%$wU/
s=@(PB`P<D}/D_W1MP@>C
.^( Ńi%؛.
ZPgT y/PY@{p^;s.Y,?a>,pb1,EQ7p^.%a08Zd]P/^px@*eHPr
in5qRQ!%7jaqi%Py$@+ /9@PP9šg
`(4&ЂR
<6YJ@(T8C0v`$\>@^R#*H A7iB68))0ux
`$`g"@
v'B*P`7gU@BrZY+p{rJ kA @=Ah`H&". Ip ǌ ̠&LPЕM`0JGvW
&`Uc@@( C#[B`6TT
XhMr>+%6B"@H
;XBQc*1Α PnXjԩBI@(@4P_.8]@ 6~Pil 5Hhbh>Zԧ$@*0Ȅ?L
$!:)=xpFF@i@<Ёm@rAـG^lgGX"PiM(pe]Ps
a@<P)@< CHE07b
ZVpY
!HA
0`0A"jbp!RS
>07[pe Hp t)G5dV2I˦b4n4>bL*$L7p#4&@)Fur8i5@M]BP0@N &xr.M@AOE?`ig)04<0wRhrA1P25U7/dIpu10Gw*_0S?h+=$+M8@0d^`` $ph@4uQ@
`72.B Idr#i5mQ,6'vx!y$TfI)c*!02%`4
(Jr4'e3 H
Z{;Pl
ug0gA iQg45 P~UN&Jp
50"YprWM7Rd`[rFgK0csJ>L`/cLg Uiŏ#pnjpܳ'mQ%=Qp.:ԡ0y4@&S3PP7:ΦI [EY$GN+:q<8B%(`a 1@5G030268"@Im7hPQ"6 ? +{\G@.PP
BC>+(*&hաII+ 0`G4#8 VN5, 80ax>1$`_@7`=0090<3&:ߴhPh3@= HrCS1)P:Pm/&=x=av0G I*`%Q+Fp&6ޣs!Pcvb*h6w`S\&'@::?ge`]x!P@/0K #+Q!\? ]6X1$L H`*7@9@Jb +/ H`s\AC0F
9.y9GP`CO
521 ZuAJ 14u`"Ԉ%P}
ӄS
@be)S`!((@.Zi7IsdQ(> `?6<3(pO0hE5ftƚ30/
gQDkxi
@<@! OD `"BZ!`X c1J@
5<0QQA$C$> +16FL0p`639_Js͆*@(P>vA4>c% p2v&VR83;Q5&x%P@8')&p+QNbQ~t !)@/o)qՁ˂6I0+ M8=@S;IP`A+G@Z90C 15`qQGp5#`aRP?n
8%\c{I=[; @>= 09Py.,pB@7v4
1=4)!@0
2p)zy!
(^,0!5@&ř!]** o;:Y=) KCH0,
*b*'J7P!:Y@[?g;PR`j060Up+pj+PWD\xA)^)4M(fZ<*c6d.rĚdHG@S/<006,"wA@ B`i:(@0J/PE94D0# "677+ϴ%9G!f>!<`$Up'F maȭn@<9B@*PVc,5eSn*@!5! P
R06La@@}JG, R Pk/0{ 7 A%n
fXE?`9*3`7PU@YP38` M5pL8LP5P7 b /,sdE`'Rz@yx)#r@V#&'yIppA^<5@* ~)lHH"`= RYCRZ<()puԽ_xQ#`'$:CPb2qP07K@ My,e_ 3.abS42ƬT0QnQa0I{Xrہn3A#@'`oQKR(l2pքy /1SB51,=Lk
)`2S4@51.@#ZT00dc0hcO3(5335(Sm_0L d9@:.p=@G1Sl#E8mAw*^ VdQ3 tx'`y"3p+X[YXB@(FN0e;()H4Iko@ZO00F`~k[Q0N0ZH6@LXP ,Q.Ca $0#![%Uų]E'5(!Sφ. @p5Qћ ]0r@`a$NXѢDNPC*$0 $X (\h1!0a "l0atP6<!Ă F!
9X&6(P,Z(r
\fؘ@Cn!n( ,`8"CcvlhA#&C^ܐRbF8zpaK.`LhD8 `q*0Pp"Wq]
(PAT~6RHhC&bh pH@p聄$hhn~ p xp`thX`pAV3#h>,(pb
l
` rhXaR 0A(Р !80AC V@0",@"J"<`6` @XHUU
A"K)"0.&#Kn1~AvA
#6`1R`B eȢ@ 'p XvHbv(@%rPp؊
rE&((`
>0Pb>0` F0p 0 B&*(@ P8tU@Z@RBA(
XhhH j[B(@> (6XH,Pak=J#BdpC
OG;>PA,808$6Xjj<
" l0MPoԀPgzBP*0w
!exK\$(@+0&ug9V
PzbnV<@ 3f!%N[@msJO54g`'@#PT)`;X`&*,`2qQ, "Y(iQf
E"`: '17\&TG_4JJVp400CA`.09G2)i&U8 b`rP (_`_@)PO (+5_Q/`(E%0:0R["/"@0C8Q ȅ%@0Pb;`F:RG%,0@l@`&h0uD$lL5s@b.$P
:D )%.Pރ&âl<A/@J<;#*!2.@KPk'P6p+p#C;0Yu")EDS@`y9=p1@Jaid]&p}AA22d77_p
P.Mx`8;F,p;ePG* Zuye'2{P
a
Q;b2@a.p@rPc=q41`t88Qig`C`d23C60I?DP`3`CPbP1U#I?VpYLJLPE3)s2L@S>w#, qH44u%P4"eP*p6(B[a6HVu IE@v bD:MT$=#@iHB)U Y k7 F 2!ME0/`!3Qj$0.P~dNL2;C@KJnH@wzU
2,>40P{@?7~gRQ.,+P:h)P&pp
q{ha?/ xApCA<Ѓ:uQ@%{~b]z/PmT4#r3M70> (0vD.(+NzH>#+va(X%/=+6pqXj6Xix`M[`1[i;tl#@G`ttb"r4r176s= +~NR p䘜'7NX,<@E{w*"P6;#i!11MS3H
B_bPu95% 0E0'@sZ$`I@
`>u? PO$pV;S'g1,*PAidrl2AT: @"P3@S1(k!&>yQI1PPqq/!
0Ϣ
.90J$((0> 16A@4p9+`H= //90[@RHc49hf08g%' ?gZƨ).:˳gMwQ;Ha"
7*E+R1H~}U(@
PsB*.1)00]bXrE#
%Ww<p<0\s 09 # `EogR46`\ໄ7"db"U `/@tggVFBKPI@yQΪ Ao#]zaMh&c"p%tc
0aFv#`
(1.29)
Hence, the work of the field forces on the charge q can be expressed
through the potential difference:
Ai 2 = W P ' t W p . x =q "
2 , and Eq. (1.45) becomes § E dl = 0 (1.46) (the circle on the integral sign indicates that integration is per formed over a closed contour). It must be noted that this relation holds only for an electrostatic field. We shall see on a later page that the field of moving charges (i.e. a field changing with time) is not a po tential one. Therefore, condition (1.46) is not observed for it. An imaginary surface all of whose points have the same potential is called an equipotential surface. Its equation has the form cp Vi 2 ) = const The potential does not change in movement along an equipotential surface over the distance dl (d
Hence
£ w£^ 1 + 3cos,e < 133 >
Assuming in Eq. (1.53) that 0 = 0, we get the strength on the
dipole axis:
Ei
1 2 p
4jteo r*
(1.54)
The vector Ey is directed along the dipole axis. This is in agreement
with the axial symmetry of the problem. Examination of Eq. (1.51)
shows that E r > 0 when 0 = 0, and E r r ~dz
1 . 100 )
which we can see to be the divergence of the vector a (see Eq. (1.81)1
Finally, the vector_product of the vectors V and a gives a vector
with the components (Val* = VyO* — V z 2 „ — dajdy — da y /dz,
etc., that coincide with the components of curl a (see Eqs. (1.92)
(1.94)1. Hence, using the writing of a vector product with the aid
50
Electricity and Magnetism
of a determinant, we have
curl a = [Va] =
d
dx
d
dy
a.,
d
dz
a z
( 1 . 101 )
Thus, there are two ways of denoting the gradient, divergence,
and curl:
V
, Va = diva, [Va]s=curla The use of the del symbol has a number of advantages. We shall therefore use such symbols in the following. One must accustom oneself to identify the symbol V9 with the words “gradient of phi” (i.e. to say not “del phi”, but “gradient of phi”), the symbol Va with the words “divergence of a” and, finally, the symbol [val with the words “curl of a”. When using the vector V, one must remember that it is a dif ferential operator acting on all the functions to the right of it. (Consequently, in transforming expressions including V, one must 'take into consideration both the rules of vector ^algebra and those of differential calculus. For example, the derivative of the product of the functions
= V ( V
(A is the Laplacian operator) curl grad 9 = [v, V9I = fwl 9 — 0 ( 1 . 105 ) (we remind our reader that the vector product of a vector and itself is zero). Let us apply the divergence and curl operations to the function curl a: div curl a = V IVal = 0 ( 1 . 106 ) Electric Field in a Vacuum 51 (a scalar triple product equals the volume oi a parallelepiped con structed on the vectors being multiplied (see Vol. I, p. 31); if two of these vectors coincide, the volume of the parallelepiped equals zero); curl curl a = lv, tVall = V (Va) — (W) a = = grad div a — Aa (1.107) [we have used Eq. (1.35) of Vol. I, p. 32, namely, [a, [bell * = b (ac ) — c (ab)J. Equation (1.106) signifies that the field of a curl has no sources. Hence, the lines of the vector lVal have neither a beginning nor an end. It is exactly for this reason that the flux of a curl through any surface S resting on the given contour T is the same [see Eq. (1.97)]. We shall note in concluding that when the del operator is used, Eqs. (1.82) and (1.97) can be given the form a • dS = j Va dV (the OstrogradskyGauss theorem) (1.408) ^ a dl =  [Va] dS (Stokes's theorem) (1.109) 1.12. Circulation and Curl of an Electrostatic Field We established in Sec. 1.6 that the forces acting on the charge q in an electrostatic field are conservative. Hence, the work of these forces on any closed path T is zero: A = ^ gE dl = 0 r Cancelling q, we get §Edl = 0 (1.110) r (compare with Eq. (1.46)]. The integral in the lefthand side of Eq. (1.110) is the circulation of the vector E around contour T [see expression (1.80)]. Thus, an electrostatic field is characterized by the fact that the circulation of the strength vector of this field around any closed contour equals zero. Let us take an arbitrary surface S resting on contour T for which the circulation is calculated (Fig. 1.32). According to Stokes’s theorem [see Eq. (1.109)], the integral of curl E taken over this 52 Electricity and Magnetism surface equals the circulation of the vector E around contour T: ( 1 . 111 ) Since the circulation equals zero, we arrive at the conclusion that  (VE]dS = 0 This condition must be observed for any surface S resting on arbitrary contour I\ This is possible only if the curl of the vector E at every point of the field equals zero: IVE) = 0 (1.112) By analogy with the fan impeller shown in Fig. 1.25, let us ima gine an electrical “impeller” in the form of a light hub with spokes E Fig. 1.32 Fig. 1.33 whose ends carry identical positive charges q (Fig. 1.33; the entire arrangement must be small in size). At the points of an electric field where curl E differs from zero, such an impeller would rotate with an acceleration that is the greater, the larger is the projection of the curl onto the impeller axis. For an electrostatic field, such an imaginary arrangement would not rotate with any orientation of its axis. Thus, a feature of an electrostatic field is that it is a noncircuital one. We established in the preceding section that the curl of the gradient of a scalar function equals zero (see expression (1.96)]. Therefore, the equality to zero of curl E at every point of a field makes it possible to represent E in the form of the gradient of a sca lar function
£ V i *
1.13. Gauss’s Theorem
rt
r ^7
A 
, *
$
' 'IV
We established in the preceding section what the curl of an elec
trostatic field equals. Now let us find the divergence of a field. For
this purpose, we shall consider the field of
a point charge q and calculate the flux of
the vector E through closed surface S sur
rounding the charge (Fig. 1.36). We showed
in Sec. 1.5 that the number of lines of the
vector E beginning at a point charge \q or
terminating at a charge — q numerically
equals q/e 0 .
By Eq. (1.77), the flux of the vector E
through any closed surface equals the num
ber of lines coming out, i.e. beginning on the
charge, if it is positive, and the number of
lines entering the surface, i.e. terminating
on the charge, if it is negative. Taking into
account that the number of lines beginning or terminating at a point
charge numerically equals g/e 0 (see Sec. 1.5), we can write that
Fig. 1.36
The sign of the flux coincides with that of the charge q. The dimen
sions of both sides of Eq. (1.113) are identical.
Now let us assume that a closed surface surrounds N point charges
q u q * t  • .. Qn On the basis of the superposition principle, the
strength E of the field set up by all the charges equals the sum of the
54
Electricity and Magnetism
strengths E t set up by each charge separately: E = 2 E Hence,
i = 23 ?t (

0, accordingly, o' for it is
positive; for the lefthand surface P n < 0,
accordingly, o' for it is negative.
Expressing P through x and E by means
of Eq. (2.5), we arrive at the formula
o' = %* 0 E n (2.10)
where E n is the normal component of the
field strength inside the dielectric. According
to Eq. (2.10), at the places where the field
lines emerge from the dielectric (E^ >0),
positive bound charges come up to the surface, while where the field
lines enter the dielectric ( E n < 0), negative surface charges appear.
Equations (2.9) and (2.10) also hold in the most general case when
an inhomogeneous dielectric of an arbitrary shape is in an inhomo
geneous electric field. By P n and E n in this case, we must understand
the normal component of the relevant vector taken in direct proxim
ity to the surface element for which o' is being determined.
Now let us turn to finding the volume density of the bound charges
appearing inside an inhomogeneous dielectric. Let us consider an
imaginary small area AS (Fig. 2.3) in an inhomogeneous isotropic
dielectric with nonpolar molecules. Assume that a unit volume
of the dielectric has n identical particles with a charge of ~\e and n
identical particles with a charge of — e. In close proximity to area
AS , the electric field and the dielectric can be considered homo
geneous. Therefore, when the field is switched on, all the positive
charges near AS will be displaced over the same distance Z x in the
direction of E, and all the negative charges will be displaced in
the opposite direction over the same distance Z 2 (see Fig. 2.3). A cer
tain number of charges of one sign (positive if a
E =
l + x c
Thus, in the given case, the permittivity e
shows how many times the field in a di
electric weakens.
Multiplying Eq. (2.33) by e 0 e, we get the
electric displacement inside the plate:
D = e 0 eE = e 0 E 0 = D 0 (2.34)
Hence, the electric displacement inside the
plate coincides with that of the external
field£> 0 . Substituting
Comparing Eqs. (3.6) and (3.5), we find that the capacitance of an isolated sphere of radius R immersed in a homogeneous infinite dielectric of permittivity e is C = 4 m 0 sR (3.7) The unit of capacitance is the capacitance of a conductor whose potential changes by 1 V when a charge of 1 C is imparted to it. This unit of capacitance is called the farad (F). In the Gaussian system, the formula for the capacitance of an isolated sphere has the form C = eR (3.8) Since e is a dimensionless quantity, the capacitance determined by Eq. (3.8) has the dimension of length. The unit of capacitance is the capacitance of an isolated sphere with a radius of 1 cm in a vacuum. This unit of capacitance is called the centimetre. According to Conductors in an Electric Field 89 Eq. (3.5), 1 F = ^r = test c 8 9e c " ' 9 x 1°“ ^ (3.9) An isolated sphere having a radius of 9 X 10* m, i.e. a radius 1500 times greater than that of the Earth, would have a capacitance of 1 F. We can thus see that the farad is a very great unit. For this reason, submultiples of a farad are used in practice— the millifarad (mF), the microfarad (pF), the nanofarad (nF), and the picofarad (pF)'(see Vol. I, Table 3.1, p. 82). 3.4. Capacitors Isolated conductors have a small capacitance. Even a sphere of the Earth’s size has a capacitance of only 700 pF. Devices are needed in practice, however, that with a low potential relative to the sur rounding bodies would accumulate charges of an appreciable magni tude (i.e. would have a high charge “capacity”). Such devices, called capacitors, are based on the fact that the capacitance of a conductor grows when other bodies are brought close to it. This is due to the circumstance that under the action of the field set up by the charged conductor, induced (on a conductor) or bound (on a dielectric) char ges appear on the body brought up to it. Charges of the sign opposite to that of the charge q of the conductor will be closer to the conductor than charges of the same sign as q and, consequently, will have a greater influence on its potential. Therefore, when a body is brought close to a charged conductor, the potential of the latter diminishes in absolute value. According to Eq. (3.5), this signifies an increase in the capacitance of the conductor. Capacitors are made in the form of two conductors placed close to each other. The conductors forming a capacitor are called its plates. To prevent external bodies from influencing the capacitance of a capacitor, the plates are shaped and arranged relative to each other so that the field set up by the charges accumulating on them is con centrated inside the capacitor. This condition is satisfied (see Sec. 1.14) by two plates arranged close to each other, two coaxial cylinders, and two concentric spheres. Accordingly, parallelplate (plane), cylindrical, and spherical capacitors are encountered. Since the field is confined inside a capacitor, the electric displacement lines begin on one plate and terminate on the other. Consequently, the extraneous charges produced on the plates have the same magni tude and are opposite in sign. The basic characteristic of a capacitor is its capacitance, by which is meant a quantity proportional to the charge q and inversely pro 90 Electricity and Magnetism portional to the potential difference between the plates: ( 3 . The potential difference
e = permittivity of the substance filling the gap.
It must be noted that the accuracy of determining the capacitance
of a real parallelplate capacitor by Eq. (3.12) is the greater, the
smaller is the separation distance d in comparison with the linear
dimensions of the plates.
It can be seen from Eq. (3.12) that the dimension of the electric
constant e 0 equals the dimension of capacitance divided by that of
length. Accordingly, e 0 is measured in farads per metre [see
Eq. (1.12)].
* A more general definition of the quantity called voltage will be given in
Sec. 5.3 [see Eq. (5.18)].
Conductors in an Electric Field
91
If we disregard the dispersion of the field near the plate edges, we
can easily obtain the following equation for the capacitance of
a cylindrical capacitor:
r 2jte<,eZ
~ In
(3.13)
where l = length of the capacitor
and R 2 = radii of the internal and external plates.
The accuracy of determining the capacitance of a real capacitor
by Eq. (3.13) is the greater, the smaller is the separation distance
of the plates d = R 2 — R 2 in comparison with l and R v
The capacitance of a spherical capacitor is
< 314 >
where and fi 2 are the radii of the internal and external plates.
Apart from the capacitance, every capacitor is characterized by
the maximum voltage t/ max that may be applied across its plates
without the danger of a breakdown. When this voltage is exceeded,
a spark jumps across the space between the plates. The result is
destruction of the dielectric and failure of the capacitor.
CHAPTER 4 ENERGY OF
AN ELECTRIC FIELD
4.1. Energy of a Charged Conductor
The charge q on a conductor can be considered as a system of point
charges A q. In Sec. 1.7, we obtained the following expression for the
energy of interaction of a system of charges (see Eq. (1.39)1:
W v = ( 4  1 )
Here 2 ) + ?£« (5.17)
1 1
The quantity numerically equal to the work done by the electrostat
ic and extraneous forces in moving a unit positive charge is defined
as the voltage drop or simply the voltage U on the given section of
the circuit. According to Eq. (5.17),
— 92+^12 (5.18)
A section of a circuit on which no extraneous forces act is called
homogeneous. A section on which the current carriers experience
extraneous forces is called inhomogeneous. For a homogeneous sec
104
Electricity and Magnetism
tion of a circuit
=
0. If the e.m.f. prevents the motion of the positive carriers in the given direc tion, t l2 < 0. Let us write Eq. (5.27) in the form /== il _?»+ j L , (5.28) This equation expresses Ohm’s law for an inhomogeneous circuit section. Assuming that q) x =
4 —    £r 2 < r *> ( 7  5 °)
(Z is the atomic number of a chemical element; the number of elec
trons in an atom is Z).
Thus, the action of an external magnetic field sets up precession
of the electron orbits with the same angular velocity (7.46) for all
the electrons. The additional motion of the electrons due to preces
sion leads to the production of an induced magnetic moment of an
atom {Eq. (7.50)] directed against the field. Larmor precession ap
pears in all substances without exception. When atoms by themselves
have a magnetic moment, however, a magnetic field not only induces
the moment (7.50), but also has an orienting action on the magnetic
moments of atoms, aligning them in the direction of the field. The
positive (i.e. directed along the field) magnetic moment that appears
may be considerably greater than the negative induced moment.
The resultant moment is therefore positive, and the substance be
haves like a paramagnetic.
Diamagnetism is found only in substances whose atoms have no
magnetic moment (the vector sum of the orbital and spin magnetic
moments of the atom electrons is zero). If we multiply Eq. (7.50)
by the Avogadro constant N A for such a substance, we get the mag
netic moment for a mole of the substance. Dividing it by the field
strength H y we find the molar magnetic susceptibility Xm.moi
The permeability of dielectrics virtually equals unity. We can there
fore assume that BtH = p 0 . Thus,
174
Electricity and Magnetism
We must note that the strict quantummechanical theory gives
exactly the same expression.
Introduction of the numerical values of p 0 , N A , *, and m in
Eq. (7.51) yields
z
Xra, mol = — 3.55 X 10® 2 ( r t)
The radii of electron orbits have a value of the order of 10~ 10 m.
Hence, the molar diamagnetic susceptibility of the order of 10“ lt 10~ , 9
is obtained, which agrees quite well with experimental data.
7.8. Paramagnetism
If the magnetic moment p m of the atoms differs from zero, the
relevant substance is paramagnetic. A magnetic field tends to align
the magnetic moments of the atoms along B, while thermal motion
tends to scatter them uniformly in all directions. As a result, a cer
tain preferential orientation of the moments is established along
the field. Its value grows with increasing B and diminishes with
increasing temperature.
The French physicist and chemist Pierre Curie (18591906) estab
lished experimentally a law (named Curie’s law in his honour)
according to which the susceptibility of a paramagnetic is
Xm.mol = f (7.52)
where C — Curie constant depending on the kind of substance
T = absolute temperature.
The classical theory of paramagnetism was developed by the
French physicist Paul Langevin (18721946) in 1905. We shall limit
ourselves to a treatment of this theory for not too strong fields and
not very low temperatures.
According to Eq. (6.76), an atom in a magnetic field has the
potential energy W — — p m B cos 0 that depends on the angle 0
between the vectors p m and B. Therefore, the equilibrium distribu
tion of the moments by directions must obey Boltzmann’s law (see
Sec. 11.8 of Vol. I, p. 326 et seq.). According to this law, the prob
ability of the fact that the magnetic moment of an atom will make
with the direction of the vector B an angle within the limits from
0 to 0 + d0 is proportional to
( W \ i p m B cos 0 \
TEf) = ,x P( «■ )
Introducing the notation
(7.53)
Magnetic Field in a Substance
175
W e can write the expression determining the probability in the form
\ exp (a cos 0) (7.54)
In the absence of a field, all the directions of the magnetic moments
are equally probable. Consequently, the probability of the fact that
the direction of a moment will form with a certain direction z an
angle within the limits from 0 to 0 + d0 is
/JD \ d ®o 2ji sin 0 dQ i . A , A
(dPo)B = 0 = ~4^ = 4^ = “2sm 0d0 ( / .55)
Here dQ e = 2n sin 0 d0 is the solid angle enclosed between cones
having apex angles of 0 and 0 + <^0
(Fig. 7.16).
When a field is present, the multiplier
(7.54) appears in the expression for the
probability:
dP$ = A exp (a cos 0) sin 0 d0 (7.56)
(A is a proportionality constant that is
meanwhile unknown).
The magnetic moment of an atom has
a magnitude of the order of one Bohr
magneton, i.e. about 10~ 2S J/T [see Eq. (7.45)]. At the usually
achieved fields, the magnetic induction is of the order of 1 T (10 4 Gs).
Hence, p m B is of the order of 10~ 23 J. The quantity kT at room tem
perature is about 4x 10“ 21 J. Thus, a — p m B/kT is much smaller
than unity, and exp ( a cos 0) may be replaced with the approximate
expression 1 + a cos 0. In this approximation, Eq. (7.56) becomes
dPe = A (1 + a cos 0) sin 0 dQ
The constant A can be found by proceeding from the fact that
the sum of the probabilities of all possible values of the angle 0
must equal unity:
ji
1= J A(l + a cos 0) sin 0 dQ == A
o
Hence, A = 1, so that
dPe = y (1 + a cos 0) sin 0 d0
Assume that unit volume of a paramagnetic contains n atoms.
Consequently, the number of atoms whose magnetic moments form
angles from 0 to 0 4 dQ with the direction of the field will be
dne = n dP e = ^ n (1 + a cos 0) sin 0 d0
Fig. 7.16
170
Electricity and Magnetism
Each of these atoms makes a contribution of p m cos 0 to the resultant
magnetic moment. Therefore, we get the following expression for
the magnetic moment of unit volume (i.e. for the magnetization):
n
M = j p m cos 0 dn 0 —
o
K
= Y^p m j (1 + a cos 6) cos 0 sin 0 d0 = np m
o
Substitution for a of its value from Eq. (7.53) yields
M
n P\oP
3kT
Finally, dividing M by H and assuming that BlH = p 0 (for a para
magnetic (i is virtually equal to unity), we find the susceptibility
Xm =
Po"P&>
3*r
(7.57)
Substituting the Avogadro constant for n, we get an expresion
for the molar susceptibility:
Xm, mol :
3kT
(7.58)
We have arrived at Curie’s law. A comparison of Eqs. (7.52) and
(7.58) gives the following expression for the Curie constant:
3k
(7.59)
It must be remembered that Eq. (7.58) has been obtained assuming
that p m B <£LkT. In very strong fields and at low temperatures,
deviations are observed from proportionality between the
magnetization of a paramagnetic M and the field strength H . In
particular, a state of magnetic saturation may set in when all the
p m *s are lined up along the field, and a further increase in H does
not result in a growth in M .
The values of Xm, moi calculated by Eq. (7.58) in a number of
cases agree quite well with the values obtained experimentally.
The quantum theory of paramagnetism takes account of the fact
that only discrete orientations of the magnetic moment of an atom
relative to a field are possible. It arrives at an expression for Xm»ruoi
similar to Eq. (7.58).
Magnetic Field in a Substance
177
7.9. Ferromagnetism
Substances capable of having magnetization in the absence of
an external magnetic field form a special class of magnetics. Accord
ing to the name of their most widespread representative — ferrum
(iron) — they hSive been called ferromagnetics. In addition to iron,
they include nickel, cobalt, gadolinium, their alloys and compounds,
and also certain alloys and compounds of manganese and chromium
Fig. 7.18
with nonferromagnetic elements. All these substances display fer
romagnetism only in the crystalline state.
Ferromagnetics are strongly magnetic substances. Their magneti
zation exceeds that of dia and paramagnetics which belong to the
category of weakly magnetized substances an enormous number of
times (up to 10 10 ).
The magnetization of weakly magnetized substances varies lin
early with the field strength. The magnetization of ferromagnetics
depends on H in an intricate way. Figure 7.17 shows the magnetiza
tion curve for a ferromagnetic whose magnetic moment was ini
tially zero (it is called the initial or zero magnetization curve). Already
in fields of the order of several oersteds (about 100 A/m), the magne
tization M reaches saturation. The initial magnetization curve in
a BH diagram is shown in Fig. 7.18 (curve 01). We remind our
reader that B = p 0 (H + M). Therefore, when saturation is reached,
B continues to grow with increasing H according to a linear law:
B — p 0 £f + const, where const = p 0 ^sat*
A magnetization curve for iron was first obtained and investigat
ed in detail by the Russian scientist Aleksandr Stoletov (1839
1896). The ballistic method of measuring the magnetic induction
which he developed has been finding wide application (see Sec. 8.3).
Apart from the nonlinear relation between H and M (or between
H and B), ferromagnetics are characterized by the presence of hyster
esis. If we bring magnetization up to saturation (point 1 in Fig. 7.18)
178
Electricity and Magnetism
and then diminish the magnetic held strength, the induction B
will no longer follow the initial curve 01 , but will change in accor
dance with curve 12 . As a result, when the strength of the external
held vanishes (point 2), the magnetization does not vanish and is
characterized by the quantity B v called the residual induction.
The magnetization for this point has the value Af r called the reten
tivity or remanence.
The magnetization vanishes only under the action of the held
H c directed oppositely to the held that produced the magnetization.
The held strength H c is called the coer
cive force.
The existence of remanence makes it
possible to manufacture permanent mag
nets, i.e. bodies that have a magnetic mo
ment and produce a magnetic held in the
space surrounding them without the
expenditure of energy for maintaining the
macroscopic currents. A permanent mag
net retains its properties better when the
coercive force of the material it is made
of is higher.
When an alternating magnetic held acts
on a ferromagnetic, the induction changes
in accordance with curve 123451
(Fig. 7.18) called a hysteresis loop (a simi
lar curve is obtained in an MH dia
gram). If the maximum values of H are
such that the magnetization reaches satu
ration, we get the socalled maximum hysteresis loop (the solid loop
in Fig. 7.18). If saturation is not reached at the amplitude values of
H y we get a loop called a partial cycle (the dash line in the figure).
The number of such partial cycles is inhnite, and all of them are
within the maximum hysteresis loop.
Hysteresis results in the fact that the magnetization of a ferromag
netic is not a unique function of H. It depends very greatly on the
previous history of a specimen — on the fields which it was in pre
viously. For example, in'a field of strength H 1 (Fig. 7.18), the induc
tion may have any value ranging from B[ to B \ .
It follows from everything said above about ferromagnetics that
they are very similar in their properties to ferroelectrics (see Sec. 2.9).
In connection with the ambiguity of the dependence of B on H f
the concept of permeability is applied only to the initial magnetiza
tion curve. The permeability of ferromagnetics p (and, consequently,
their magnetic susceptibility Xm) is a function of the field strength.
Figure 7.19a shows an initial magnetization curve. Let us draw
from the origin of coordinates a straight line that passes through an
Magnetic Field in a Substance
179
arbitrary point on the curve. The slope of this line is proportional
to the ratio B/H, i.e. to the permeability p for the relevant value
of the field strength. When H grows from zero, the slope (an = — B dS .
Hence, Eq. (6.86) can also be written in the form of Eq. (6.85).
The quantity d
/dt ), where
L ~ 2 nr 0 ~dt
The magnetic field is perpendicular to the plane of the orbit. We
can therefore assume that O = nrj {B), where (B) is the average
value of the magnetic induction over the area of the orbit. Hence,
Let us write the relativistic equation of motion of an electron in
orbit:
£( ~»v ) =eE+ ‘ i,B ’ rt
0 its value
from Eq. (13.3), we get
C/ m =]/ ~Im (13.9)
We can also obtain this equation if we proceed from the fact that the
maximum value of the energy of the electric field y C Um must equal
the maximum value of the energy of the magnetic field LI\
13.3. Free Damped Oscillations
Any real circuit has a resistance. The energy stored in the circuit
is gradually spent in this resistance for heating, owing to which the
free oscillations become damped. Equation
(5.27) written for circuit 132 shown in Fig. 13.3
has the form
1R= — ^ — L (13.10)
[compare with Eq. (13.1)1. Dividing this equa
• • •
tion by L and substituting q for I and q for
dl/dt, we obtain
'i+irh+Tc^ 0 ( 1311 >
Taking into account that the reciprocal of LC equals the square of
the natural frequency of the circuit o> 0 [see Eq. (13.3)1, and introduc
ing the symbol
v=4r < 1S12 >
Eq. (13.11) can be written in the form
q + 2p? + w\q — 0 (13.13)
This equation coincides with the differential equation of damped
mechanical oscillations [see Eq. (7.11) of Vol. I, p. 1891.
When <
for the phase angle.
268
Electricity and Magnetism
The product IR equals the voltage U R across the resistance, q!C is
the voltage across the capacitor U c , and the expression L ( dlldt ) de
termines the voltage across the inductance U L . Taking this into
account, we can write
U R + U c + U l = U m cos c ot (13.35)
Thus, the sum of the voltages across the separate elements of a cir
cuit at each moment of time equals the voltage applied from an ex
ternal source (see Fig. 13.5).
According to Eq. (13.31)
U R = RI m cos (
C R (13.48) When
in the denominator of Eq. (13.47) is called the Impedance, If a circuit consists only of a resistance i?, the equation of Ohm’s law has the form IR = U m cos of Hence it follows that the current in this case varies in phase with the voltage, while the amplitude of the current is A comparison of this expression with Eq. (13.47) shows that the replacement of a capacitor with 'a shorted circuit section signifies a transition to C = oo instead of to C ~ 0. Any real circuit has finite values of R y L , and C. It may happen that some of these parameters are such that their influence on the 272 Electricity and Magnetism current may be disregarded. Suppose that R of a circuit may be assumed equal to zero, and C equal to infinity. Now, we can see from Eqs. (13.47) and (13.48) that / = Um Im <&L (13.50) and that tan 9 » 00 (accordingly, 9 = n/2). The quantity X L = d)L (13.51) is called the inductive reactance. If L is expressed in henries, and co in rad/s, then X L will be expressed in ohms. Examination of Eq. (13.51) shows that the inductive reactance grows with the fre quency co. An inductance does not react to a steady current (to = 0), i.e. X L = 0. The current in an inductance lags behind the voltage by n/2. Accordingly, the voltage across the inductance leads the current by n/2 (see Fig. 13.6). Now let us assume that R and L both equal zero. Hence, according to Eqs. (13.47) and (13.48), we have tan 9 /  m i/mC 00 (i.e» 9 = — n/ 2 ). The quantity (13.52) (13.53) is called the capacitive reactance. If C is expressed in farads, and co in rad/s, then Xc will be expressed in ohms. It follows from Eq. (13.53) that the capacitive reactance diminishes with increasing frequency. For a steady current, X c = 00 — a steady current cannot flow through a capacitor. Since 9 = — n/2, the current flowing through a capacitor leads the voltage by n/2. Accordingly, the vol tage across a capacitor lags behind the current by n/2 (see Fig. 13.6). Finally, suppose that we may assume R to equal zero. In this case, Eq. (13.47) becomes The quantity / m = u m 
(13.63) The factor cos
0. Hence, Eq. (14.4)
describes a wave propagating in the direction of growing x . A wave
propagating in the opposite direction is described by the equation
6 — ,4cos[©(* + ^)+a] (14.7)
Indeed, equating the phase of wave (14.7) to a constant and differen
tiating the equation obtained , we arrive at the expression
dx
280
Waves
from which it follows that the wave given by Eq. (14.7) propagates
in the direction of diminishing x .
The equation of a plane wave can be given a symmetrical form
relative to x and U For this purpose, let us introduce the quantity
(14.8)
known as the wave number. Multiplying the numerator and the de
nominator of Eq. (14.8) by the frequency v, we can represent the
wave number in the form
(14.9)
[see Eq. (14.2)]. Opening the parentheses in Eq. (14.4) and taking
Eq. (14.9) into account, we arrive at the following equation for a
plane wave propagating along the xaxis:
 = A cos (c ot — kx + a) (14*10)
The equation of a wave propagating in the direction of diminishing x
differs from Eq. (14.10) only in the sign of the term kx.
In deriving Eq. (14.10), we assumed that the amplitude of the
oscillations does not depend on x. This is observed for a plane wave
when the energy of the wave is not absorbed by the medium. When
a wave propagates in a medium absorbing energy, the intensity of
the wave gradually diminishes with an increasing distance from the
source of oscillations — damping of the wave is observed. Experi
ments show that in a homogeneous medium such damping occurs
according to an exponential law: A = A 0 e~v x [compare with the
diminishing of the amplitude of damped oscillations with time;
see Eq. (7.102) of Vol. I, p. 210]. Accordingly, the equation of a
plane wave has the following form:
£ = A 0 ev*cos (

LV dx /x+A 3C+S+&6 '
l dx Jx+ZJ
(14.32)
The value of the derivative 5£/<9x in the section x 4* 6 can be
written with great accuracy for small values of 6 in the form
(#U(£).+[w(4)].M4),+&« < 14  33 »
where by
(the relative elongation d\ldx in elastic deformations is much smal
ler than unity. Consequently, A£  (16.7)
Averaging is performed over the time of “operation” of the instru
ment, which, as we have already noted, is much greater than the
period of oscillations of the wave. The intensity is measured either
in energy units (for example, in W/m 2 ), or in light units named “lu
men per square metre” (see Sec. 16.5).
According to Eq. (15.22), the magnitudes of the amplitudes of
the vectors E and H in an electromagnetic wave are related by the
expression
Em V ee 0 = Hm V HPo — Hm V Po
(we have assumed that p = 1). It thus follows that
where n is the refractive index of the medium in which the wave
propagates. Thus, H m is proportional to E m and n :
H m oc nE m (16.8)
The magnitude of the average value of the Poynting vector is pro
portional to E m H m . We can therefore write that
I oc nEm = nA z  (16.9)
(the constant of proportionality is ty^o/iio). Hence, the light inten
CM
sity is proportional to the refractive index of the medium and the
square of the light wave amplitude.
We must note that when considering the propagation of light in a
homogeneous medium, we may assume that the intensity is propor
tional to the square of the light wave amplitude
I oc A 2 (16.10)
For light passing through the interface between media, however, the
expression for the intensity, which does not take the factor n into
account, leads to nonconservation of the light flux.
The lines along which light energy propagates are called rays.
The averaged Poynting vector (S) is directed at each point along a
tangent to a ray. The direction of (S) in isotropic media coincides
with a normal to the wave surface, i.e. with the direction of the
wave vector k. Hence, the rays are perpendicular to the wave sur
faces. In anisotropic media, a normal to the wave surface generally
does not coincide with the direction of the Poynting vector so that
the rays are not orthogonal to the wave surfaces.
General
321
Although light waves are transverse, they usually do not dis
play asymmetry relative to a ray. The explanation is that in natural
light (i.e. in light emitted by conventional sources) there are oscil
lations that occur in the most diverse directions perpendicular to a
ray (Fig. 16.1). The radiation of a luminous body consists of the
waves emitted by its atoms. The process of radiation in an indi
vidual atom continues about 10^® s. During this
time, a sequence of crests and troughs (or, as is
said, a wave train) of about three metres in length
is formed. The atom “dies out”, and then “flares
up” again after a certain time elapses. Many atoms
“flare up” at the same time. The wave trains they
emit are superposed on one another and form the Fi S*
light wave emitted by the relevant body. The
plane of oscillations is oriented randomly for each wave train.
Therefore, the resultant wave contains oscillations of different direc
tions with an equal probability.
In natural light, the oscillations in different directions follow
one another rapidly and without any order. Light in which the direc
tion of the oscillations has been brought into order in some way or
other is called polarized. If the oscillations of the light vector occur
only in a single plane passing through a ray, the light is called plane
(or linearly) polarized. The order may consist in that the vector E
rotates about a ray while simultaneously pulsating in magnitude.
The result is that the tip of the vector E describes an ellipse. Such
light is called elliptically polarized. If the tip of the vector E describes
a circle, the light is called circularly polarized.
We shall deal with natural light in Chapters 17 and 18. For this
reason, we shall display no interest in the direction of the light
vector oscillations. The ways of obtaining polarized light and its
properties are considered in Chap. 19,
16.2. Representation of Harmonic Functions
Using Exponents
Let us form the sum of two complex numbers z x = x x + iy t and
z t =x 2 + iy 2 :
% = z v + z x a + iy X ) + (*2 + Wi) =
= (*i + x 2 ) + i (y t + y 2 ) (16.11)
It can be seen from Eq. (16,11) that the real part of the sum of com
plex numbers equals the sum of the real parts of the addends:
Re {(z t + z 2 )} = Re {zj + Re {z t } (16.12)
322 Optics
Let us assume that a complex number is a function of a certain
parameter, for example, of the time t:
z(t) = x ( t ) + iy (t)
Differentiating this function with respect to t, we get
dz dx , . dy
~dt ~dt ' 1 It
It thus follows that the real part of the derivative of z with respect
to t equals the derivative of the real part of z with respect to t:
M£H4 Re < z > < 16  13 >
A similar relation holds upon integration of a complex function.
Indeed,
 z (t) dt = £ x(t) dt + i j y (t) dt
whence it can be seen that the real part of the integral of z (0 equals
the integral of the real part of z ( t ):
Re { j z (0 dt) = j Re (z (t) dt} (16.14)
It is evident that relations similar to Eqs. (16.12), (16.13), and
(16.14) also hold for the imaginary parts of complex functions.
It follows from the above that when the operations of addition,
differentiation, and integration are performed with complex functions,
and also linear combinations of these operations, the real (imaginary)
part of the result coincides with the result that would be obtained
when similar operations are performed with the real (imaginary)
parts of the same functions*. Using the symbol L to denote a linear
combination of the operations listed above, we can write:
Re {L (s lf s 2 , ...)} = L (Re {z 4 }, R e{z 2 }, ...) (16.15)
The property of linear operations we have established makes it
possible to use the following procedure in calculations: when per
forming linear operations with harmonic functions of the form
A cos ((of — kxX — k v y — k z z + a), we can replace these functions
with the exponents
A exp [ i (cot— k x x — k v y — k z z + a) ] = A exp [ i (cof — k x x — k v y — k z z) 1
(16.16)
where A = Ae ia is a complex number called the complex amplitude*
With such representation, we can add functions, differentiate them
* We must note that this rule cannot be applied to nonlinear operations,
for example, to the multiplication of functions and squaring them.
General
323
with respect to the variables t , x, y, z, and also integrate over these
variables. In performing the calculations, we must take the real
part of the result obtained. The expediency of this procedure is
explained by the fact that calculations with exponents are consid
erably simpler than calculations performed with trigonometric
functions.
Passing over to representation (16.16), we in essence add to all
functions of the kind A cos (oof — k^x — k^y — k z z + a) the ad
dends iA sin (of — k^x — k y y — k z z f a). We remind our reader
that we have used a similar procedure when studying forced oscil
lations (see Sec. 7.12 of Vol. I, p. 215 et seq.).
16.3. Reflection and Refraction
of a Plane Wave at the Interface
Between Two Dielectrics
Assume that a plane electromagnetic wave falls on the plane inter
face between two homogeneous and isotropic dielectrics. The dielec
tric in which the incident wave is propagating is characterized by
the permittivity e lt and the second dielectric by the permittivity e a .
We assume that the permeabilities are unity. Experiments show that
in this case, apart from the plane refracted wave propagating in the
second dielectric, a plane reflected wave propagating in the first
dielectric is produced.
Let us determine the direction of propagation of the incident
wave with the aid of the wave vector k, of the reflected wave with the
aid of the vector k' and, finally, of the refracted wave with the aid
of the vector k". We shall find how the directions of k' and k" are
related to the direction of k. We can do this by taking advantage of
the fact that the following condition must be observed at the inter
face between the two dielectrics:
E ux = (16.17)
Here E Xs z and E,, x are the tangential components of the electric
field strength in the first and second medium, respectively.
In Sec. 2.7, we proved Eq. (16.17) for electrostatic fields [see
Eq. (2.44)]. It can easily be extended, however, to timevarying
fields. According to Eq. (9.5), the circulation of E determined by
Eq. (2.42) for varying fields must be not zero, but equal to the integ
ral j (B)dS taken over the area of the loop shown in Fig. 2.9:
Etdl=E Ux a— E 2 ' X a + (E b )2b= — j BdS
324
Optica
Since B is finite, in the limit transition b *• 0 the integral in the
righthand side vanishes, and we arrive at condition (2.43), from
which follows Eq. (2.44).
Assume that the vector k determining the direction of propagation
of the incident wave is in the plane of the drawing (Fig. 16.2). The
direction of a normal to the interface will be characterized by the
vector n. The plane in which the vectors k and n are is called the
plane of incidence of the wave. Let us take
the line of intersection of the plane of inci
dence with the interface between the dielec
trics as the xaxis. We shall direct the yaxis
at right angles to the plane of the dielectric
interface. The zaxis will therefore be perpen
dicular to the plane of incidence, while the
vector t will be directed along the xaxis
(see Fig. 16.2).
It is obvious from considerations of sym
metry that the vectors k' and k # can only
be in the plane of incidence (the media are
homogeneous and isotropic). Indeed, assume
that the vector k' has deflected from this
plane “toward us”. There are no grounds, however, to give such a
deflection priority over an equal deflection “away from us”. Conse
quently, the only possible direction of k' is that in the plane of
incidence. Similar reasoning also holds for the vector k*.
Let us separate from a naturally falling ray a planepolarized com
ponent in which the direction of oscillations of the vector E makes
an arbitrary angle with the plane of incidence. The oscillations of
the vector E in the plane electromagnetic wave propagating in the
direction of the vector k are described by the function*
E = E m exp li (at — kr)] = E m exp [i (c atk^x — k y y)l
Fig. 16.2
(with our choice of the coordinate axes, the projection of the vector
k onto the zaxis is zero, therefore the addend — k z z is absent in the
exponent). By correspondingly choosing the beginning of reading t ,
we have made the initial phase of the wave equal zero.
The field strengths in the reflected and refracted waves are deter
mined by similar expressions
E' = Em exp [i (co'f — k' x x — /^y + a')]
E" — Em exp [i (to”t — k r x X’—kZy + a”)]
where a 9 and a * are the initial phases of the relevant waves.
• More exactly, the real part of this function, but we shall say simply
function for brevity’s sake.
General
325
The resultant field in the first medium is
E t = Ef E' = E m exp [i (