=0 ~Hd~s I i r ≤ r 2 R = a i r + r ∇· =0; ∆ · = ρ ~ ~B ~ ~D ∇× = ; ∇× =0 ~j ~ ~H ~ ~E

!"!#$&' (')*'+

!"!#$%&()!*&+,!--./-!#(0$1!$%2"3!$ 4)&)#4(!$5&--6

∇ × ~ H ~ = ~j; ∇ × ~ E ~ = 0

∇ · ~ B ~ = 0; ∆ ~ · D ~ = ρ

^789:;

<&=#)>?$$!$ %2"!#$%&(!/!@=!)"#!$3#!5!-3A!")!#-0$1!$ B!"!($!),!"3!$9

C!#4D#!-&043!"*&1$!)@4)&)#>6E#$14D0-!7F@"04;

!!"#$%&'()*+ ,"&'-.%#/0123%-4

R = r _a + r _i

2

^7898;

52"

r ≤ r i

^1#-)6

I Hd~s ~ = 0

^789G;

,0/

r i ≤ r ≤ r a

^1'-%2

H · 2rπ = N · I; H = N · I

2π · r

^3!"45

,0/

r ≥ r a

^1'-%2

H = 0

^3!"65

!"#!$%!&' ()*+,*#-!+/)%++-,012*,*23415#!6-#))#78%+-1!+7#*25179

!!"#$%&'()(* +,'&-."/1-23#,/4,-"&/1#%// "& .52%/6724"'-&85&9'%2,."5&-&) :,!-";544.

<-'-&$-& !-'2-&=.-&3#,/4,.-4>-2,.%2-&?

≤

^(@;-AB&%2$"- -2/.'-&,&&.- C%/"5&/2-,;."5& "&

C2,'-)

!!"#$%&'()*+*,$"-)./0123##%&'$31"- %4!/%!35&$#"73&8%1"9&13:;30"-3&21<=>?).3%2#"7

1"72!/01"&$$"32909"$/#3&@/%;21;%#3&%&$$/10"&'4A0-"'3B9#%-3&4C0$3&D#/1-/3"&17#%11)

.,)$'23452)6&/7

m p

2 h v ² _p i = kT p ; → 1 eV = 1.16 · 10 ⁴

^{8&+9)' : !;<}

h v ² _p i =

=)%%+&/& >,*-/*%)24&@&24A)'-)06&)%-&/B&)+4&'2$/%&

p

^CB&)+4&'=*22&

m p

^CB&=3&/*%,/

T p

D&)6$=3+)()&/%&'@&$=&%/)&'6*''=*'-)& .&+-9&/%&)+,'0&','%&/E&/A&'-,'0-&/#$%&'()*+&

φ

^,'-

A ~

&/=)%%&+'!.F/

H ~

G.&+-&/A)/-*,44H,I0-*2D)$%GJ*9*/%24&@&2&%(*'0&A&'-&%!

!" #$%&'()*+& ,'- .&+-&'&/0)&'

!"!" #$%&'()*,%* -.'%/0)1$

φ

K20)+%

∇ × ~ E ~ = ~ 0

∇ · ~ E ~ = ρ ε

L*M&) 2&)

ε = ε ◦ · ε r ;

^=)%-&/26*+*/&'@/NO&

ε r = const.

^{: !P<}

Q&0&'-&/Q)/M&+R/&)4&)%9$'

E ~

^)2%)'-)&2&=.*++-)&D&24/&)M,'0-,/4&)'26*+*/&2#$%&'%)*+

=N0+)47

E ~ = − ∇ ~ φ

^{: !S<}

L)&T&M&'M&-)'0,'0

∇ · ~ E ~ = ρ

ε

^RF4/%-*''(,

∆φ = − ρ

ε ;

#$)22$'G@+&)4,'0 : !1<

(01'+23/

ρ = 0

456+--+72/+'+896:+; <6

∆φ = 0;

=&>-&+?@-+'*67A 4"# !;

!#$!%&' ()&*!+,-+,,BC67D%-&.67A':E/(>/67A

φ(r) = q

4πεr

^{4"# ;}

F+':+*/+/+7C67D%-&.67A+7A'-%1+A+7.+/='7+&/'%9%.+/G&H1+--?@-+'*67A+7.&($6>+/>0('?

%'07(>/'7<'>#I&(*+'J%B

φ(~x) = X

i

q i

4πε | ~x − ~x _i |

4"# ";

,3/+'7+D07%'76'+/-'*+=&.67A(K+/%+'-67AL+D0::%:&7.+7M6(./6DB

φ(~x) = 1 4πε

Z ρ (~x ^′ )

| ~x − ~x ^′ | d ³ x ^′

^{4"# N;}

!"! #$%&'(%($)*+, -%. %+%/,&(.)%$ 1%+-%.

O(A'-%L<A-#+'7+/=&.67A':+-+D%/'(*+7,+-.B

F ~ = q · E ~ W =

Z 2 1

F d~s ~ = q · (φ 2 − φ 1 )

^{4"# P;}

C0%+7<'+--+O7+/A'+ +'7+/C/0L+-&.67A

q

^{Q&6(.+:E7+7.-'*+7}

(φ = 0)

^7&*

~x

A+L/&*%B

W = q · φ(~x)

^{4"# R;}

S:,&--+'7+/D07%'76'+/-'*+7=&.67A(K+/%+'-67A':&!%&.&.,+-. A'-%.&A+A+7B

W = 1 2 Z

V

ρ(~x) · φ(~x)d ³ x

^{4"# T;}

!!"#$%&'()*+ !,-.#/0123.10& 241

N

5%&67#3$%&'0&,%18010.&%&'$01 :0#$0&01'"0)

1&/.*2%$/

1

2

&/2+3/%4)6*,4-&7 89/

N

-)42/&%& :*-,'0&')';<<! !=-*/0&4%&++%&' ;<(36+>

46&7*?@&+6&4(,A+&)6,'0B !"CD896/%!

W = X N

i =1

X

jii

q i q j

| ~x _i − ~x _j | 1 4πε = 1

2 X N

i =1

X N

j6=i

q i q j

| ~x _i − ~x _j | 1

4πε

^{B !"CD}

E)'4&%(&'-&/#$)44$'0+&)6,'089/

ρ(~x)

^)'$<)0&A+&)6,'0&/0)<%F

W = − 1 2 Z

V

ε · φ(~x) · ∆φ(~x)d ³ x

^{B !"GD}

;,446/&)<&')'2*/%!H$$/-)'*%&'I$'

∆φ

^F

φ

∂ ² φ

∂x ² + ∂ ² φ

∂y ² + ∂ ² φ

∂z ²

= ∂

∂x

φ ∂φ

∂x

− ∂φ

∂x ²

+ . . .

= ∇ · ~ (φ · ∇ ~ φ) − ∇ ~ φ · ∇ ~ φ

^{B !"JD}

1*7)%&/0)<%4)689/-)&&+&2%/)46&E'&/0)&F

W = 1 2 Z

V

ε~ ∇ φ · ∇ ~ φd ³ x − 1 2 Z

V

ε · ∇ · ~ (φ~ ∇ φ)d ³ x

^{B ! KD}

L&/46)&<&'-&4M*'-&4'*6,'-&'-+)6<(@!(,/0&46+$44&'&'?0&&/-&%&'#$%&'%)*+N36&&/0)<%F

I

φ~ ∇ φ d ~a −→ 0

−→ W = 1

2

Z ε · E ~

· E d ~ ³ x

= 1 2

Z D ~ · E d ~ ³ x

= 1

2 Z

ε E ~

2

d ³ x

^8*++4

ε

^O*2+*/)4%

B ! "D

P&)')6%)4$%/$Q&'R&-)&'2*''

ε

^{&)'S&'4$/4&)'?}

D ~

^{4%&6%-*''')6%7&6/}0/,'-43%(+)6Q*/*++&+

(,

E ~

^!

!"!#$%#$&'($%$ )*#$&+",'-**"*%( -&'..+.0$1 2&$-31!%$%1'(+*.

T7T''&/&'-&/H,0&+4&)

ε r ≥ 1

^!U)/)'%&0/)&/&'9<&/H,0&+I$+,7)'*?2$'(&'%/)46(,/:*-,'04>

I&/%&)+,'0!

4

3 R ³ π · ρ = q

^{B ! D}

.9/

r ≤ R

^F

ε · ∇ · ~ E ~ = ρ;

Z

d ³ x

^,'-A*,V46&/O*%(

→ E r = ρ ε · r

3

^{B ! WD}

!!"#$%&'()*+ ,-.-'/& '/#0$/&/1 2%'/#3-#%./&".405%%.)

,0/

r ≥ R

^1'-%2

ε ◦ · I

!"#$

E d ~a ~ = q

→ E r = q

4πε ◦ r ²

^3!"!45

6&7789-1%7&*:&;"!" "!80/.'++-+<%/'(*+=7+/1'+

W = ε

2 Z R

0

ρ ²

9ε ² r ² · 4πr ² dr + ε ◦

2 Z ∞

R

q ² · 4πr ² 16π ² ε ² _◦ r ⁴ dr

= q ²

40πεR

| {z }

%&&#'#'(&)#%$

+ q ² 8πε ◦ R

| {z }

*!+#'#'(&)#%$

3!"!>5

,0/

ε

!"#$

= ε _◦ W = 3 5

q ² 4πε ◦ R

,0/

ε

!"#$

| {z }

,-./"#$01#&# !"#$2,#'3*5#

→ ∞ : W = 1 2

q ²

4πε _◦ R

^3!"!?5

@A( .+/

1

R

B@C*D71'1<+'%E97

W

^{89-1%F.&((+( &A(}+7+/1+%'(*+7G/07.+7<+'7+H+*%+I JA7<%B -&.A711'C%KLA/M+/+*7A71.+(<-&(('(*+7

e ⁻

BN&.'A(O&*%O&7.+7@7(&%P2

QRA'E&-+7%+=7+/1'+.+/

e ⁻

NA*+O&((+ S +-+<%/'(*+,+-.+7+/1'+

6& O&7 .'+ M+(*&T+7*+'% .+(

e ⁻

^{'O U77+/+7 7'*% <+77%F (+%P% O&7 .+7}

ε

^B &C*D71'1+7 V9/8&<%9/'7.+/9C'1+7G-+'*A71PA/,+-.+7+/1'+1-+'* "6&O'%+/1'C%('*2

e ²

4πε _◦ r _e = m ◦,e · c ² r e = e ²

4πε _◦ m _◦,e · c ²

r e = 2.82 · 10 ⁻¹⁵ m;

3:-&(('(*+/

e ⁻

BN&.'A(5 3!"!W5 XT+7+,/&1+72

•

^{Y&('(%.'+H/+'7+IZ&((+.+(}

e ⁻

^9*7+,+-.[

Z+%*9.+72\=6F,+]7O&7B6'&1/&OO+3^9*/O&77F_9*+7+/1'+;*]('<5FZ&((+7/+79/O'+B

/A713^&.A71(/+79/O'+/A715

•

^{2*/,30)4%&5-/&)}6&7%$'&'8*3)+)&'9:+&;%/$'&'<=>$'&'<?*,$'&'@A 6&7%$'&'<*++&B7)'

1

2

^<1.*3)+)&'

e ⁻ , e ⁺ , ν _e , ν ¯ _e m _e c ² = 511keV e ⁻

^5%*4)+

µ ⁻ , µ ⁺ , ν µ , ¯ ν µ m µ c ² = 105.7M eV µ ⁻ , τ = 2.2µs

τ ⁻ , τ ⁺ , ν _τ , ν ¯ _τ m _τ c ² = 1784M eV τ ⁻ , τ = 3 · 10 ⁻¹³ s

^{9 ! C@}

→

D+*55)5F&G*-)&'8H/

µ ⁻ , τ ⁻

^{+*55&'5)F*'*+$0}4&5%)33&'!I

µ ⁻ , τ ⁻

^5)'-5FJ&/&K,5L 0*4&'-&5

e ⁻

^!M)&%)&8&/&N&-&,%,'0-&/-/&)6&7%$'&'8*3)+)&')5%')F%0&;+O/%!

→

^BFJ*F&2&F5&+J)/;,'057/$(&55&!

π ⁻ → µ ⁻ + ¯ ν µ ;

| {z }

!"#$%&'()*'

L

^&'+

L µ

µ ⁻ → e ⁻ + ¯ ν e + ν µ ;

| {z }

!"#$%&'()*'

L,L e ,L µ

9 ! P@

•

K;%,&++Q !"#$%&''()%**+#%'&!&R

∆m ² (ν e , ν µ ) ≥ 0

6&7%$'&'(&)0&' 4)5(,

∆ ≤ 10 ⁻¹⁸ m

F&/,'%&/;&)'&)''&/& B%/,;%,/

→

²&/-&' J&)%&/F)'

*+5:+&3&'%*/%&)+F&'4&%/*F%&%!

M)& *;%,&++& S&,%/)'$8$/5F,'0 3)% =&55,'0&' (,

∆m ²

^{,'- (,} S&,%/)'$$5()++*%)$'5L +O'0&' 8HF/% (, ;$';/&%&' :/0&4')55&'! =*' J&)T 4&/&)%5 5)F&/< -*55 &5 U$' S,++

U&/5F)&-&'&S&,%/)'$/,F&3*55&'0)4%!2*5)5%-)&V/5*F&-*8H/A

!"!# $%& '%()*+,&.* /*0+1231+*)4,%5

A ~

1.,(+234,3 /,05'43,&6)&'&(78

∇ × ~ H ~ = ~j; ∇ · ~ B ~ = 0; B ~ = µ · H ~

^9"#:$;

µ

^{()& /'<,( <,(} ()6&06=,3>+6?64,3,3 5'&,0('.(,3,(3,@63)&'3&,>)6.'34,

| µ r − 1 | ≤ 10 ⁻⁴

^23/

H ~

^{3(+&A2406B()&#}

C3)63)&,3 3(+&.(3,'0,0> 76?=.(A(,0&,0 D2)'??,3+'34

µ( H) ~

^{E (? -'.. /,)} -,006?'43,&()F

?2))=(,.&'2+/(,G604,)+(+&,,(3,H6..,9IJ)&,0,),;#

-K0

~j 6 = ~ 0

<(,&,&)(+,(3,-,./<,)+0,(<234?(&&,.)G,7&60=6&,3&('.'38

B ~ = ∇ × ~ A; ~

^{9"#: ;}

L(,),0C3)'&A,0MK..&/(,N,<,3<,/(34234

∇ · ~ B ~ = 0

^#

O(3),&A,3(3/')C?=,0,)+,1,),&A8

∇ × ~ H ~ = ~j; | · µ

∇ × ~ B ~ = µ · ~j

∇ × ~ ∇ × ~ A ~ = µ · ~j

∇ × ~ ∇ × ~ A ~ = ∇ ~ ( ∇ · ~ A) ~ − ∆ A ~ = µ · ~j

9"#:";

O(+&0'3)M60?'&(638

B ~ = ∇ × ~ A ~

^{P(0//20+,(3,@.')),Q63}R6&,3&('.,3,0MK..&8

A ~ = A ~ ^′ + ∇ ~ f ;

^/'4(.&8

∇ × ~ A ~ = ∇ × ~ A ~ ^′ ; ( ∇ × ~ ∇ ~ f = 0)

^9"#::;

f

)&,..&/'<,(,(3,<,.(,<(4,)7'.'0,-237&(63

f (~x)

^/'0

S3/,05'43,&6)&'&(7PT+.&?'3/(, !"#!$%&'()*+,"+, 8

∇ · ~ A ~ = 0

^9"#:!;

L'?(&Q,0,(3M'+&)(+/')C?=,0,)+,1,),&AA2

∆ A ~ = − µ · ~j

^9"#:U;

VW)234'3'.64A20R6())63F1.,(+234(3/,0O.,7&06)&'&(78

A(~x) = ~

Z µ · ~j(~x ^′ )

4π | ~x − ~x ^′ | d ³ x ^′

^9"#:X;

A ~

,04(<&)(+'.)623?(&&,.<'0/20+G6.2?,3(3&,40'&(63K<,0/(,)&06?MK+0,3/,3Y,0,(+,#-K0 Z/K33,[L0T+&,<,76??&?'3/'0'2)/(,\,4(3&,40'&(638

A(~x) = ~ µI 4π

Z

!"#$%&'(

d~s

| ~x − ~s |

^9"#:];

!"#$"!%&'!()*+$*)!,-".% !",!++!0)!("1!, 2+.0)#0%!"3!

!!"#$%&'()*+ ,-/0-1"'-234/0-15/#-"6-7"0 !504&$51883$"&40-&)

2)/)'%&/&33)&/&','345/-*3.&/'4&+-6-*37&)8%

b/R ≪ 1

^9:0+!;<)((&=>

x ² + y ² + z ² = R ²

~x ^′2 1 = x ² +

y + b 2

²

+ z ² = R ²

1 + b

R ² y + b ² 4R ²

~x ^′2 2 = x ² +

y − b 2

²

+ z ² = R ²

1 − b

R ² y + b ² 4R ²

9 !?@=

A ~ x

^&/0)A%3)7*,3;%/CD&'E*/*++&+(,/FGH73&>

A x ( R) = ~ µI 4πR

Z +a/2

−a/2

"

1 + by R ² + b ²

4R ^{2 − 1} 2

−

1 − by R ² + b ²

4R ^{2 − 1} 2 #

dx

^{9 !?I=}

J'%K)<&+''*7

b/R ≪ 1

^>

A _x ( R) ~ ≃ − µIab

4πR ³ · y; I · a · b =

^D*0'!L)E$+D$D&'%-&/;7+&)4&9 !MN=

H'*+$0A&<$DD%D*'>

A y ( R) ~ ≃ µIab

4πR ³ · x

^{9 !M"=}

L*D)%&/7O+%D*'45/-)&D*0'&%)37&P'-,<%)$'

B ~ = ∇ × ~ A ~

^>

B x = 3µIabxz 4πR ⁵ B y = 3µIabyz

4πR ⁵ B z = − µIab 4π

1 R ³ − 3z ²

R ⁵

= µIab

4πR ³ 3cos ² ⊖ − 1

9 !M =

!!"#$%&'()*+,%-.-"/&0"/-%&'$/123'&)4/506-760/&8"3#1"&$/-92'/!%&'/"&/-:0-621<#/"=

>/)

1(&

A ~

^{2'334'3.,(+&/,34'53#-.6))/60+,(3,-.7+,}

F ~

8,)&(44,39

φ = Z

F

B ~ · d~a = Z

F

∇ × ~ A d~a ~ =

I A d~l ~ ;

φ =

I A d~l ~

:"#;<=

>3/,02.'))()+,3?.,2&0@/A3'4(2,0)+,(3&

A ~

^'.)B(.C)4(&&,.D64

B ~

^E6,00,+3,3#

>3 /,0 F6'3&,34,+'3(2 ,02,33& 4'3 ,0)& /(, G+A)(2'.()+, H,/,6&635 /,) I,2&@0G@&,3J

E('.)

A ~

^#K'E6L(0/M,&E&/,0?(3N6))'6C/(, /,H0@5.(,JO,..,3.735,P@3

e ⁻

8,&0'+&,&#

!"!#$#%& '(%

A ~

^{)% "!*} +#,%$!%-!/,%)0

?) L,0/,3 ?QG,0(4,3&,4(& EL,( '6)/,0 5.,(+,3 R.S+2'&+@/, ,0E,65&,3

e ⁻

J%&0'+.,38,)G0@J +,3DL,.+,3'+,(3,0K0(C&)&0,2,E60T8,0.'5,06355,80'+&L,0/,3:?.,2&0@3,3(3&,0C,0@4,J

&,0=#K(,,3&)&,+,3/,3>3&,0C,0,3E46)&,0(3/,0K,&,2&@0,8,3,L,0/,34(&63/@+3,P@0+'3/,3,)

A ~

JU@&,3&('.(4%&0'+.,35'3563&,0)6+&#

K(,

e ⁻

JR,)+L(3/(52,(&),( 3(+&J0,.'&(P()&()+#K'335(.&CS0/(,/, H0@5.(,JO,..,3.735,: V";=

8,(

A(~x) = ~ ~ 0

^CS0'..,

~x

⁹

λ = h

p = h

√ 2meU

:"#;;=

?.,2&0@3,3(3&,0C,0@4,&,0:)+,4'&()+=

!!"#$%&'()*+,./012"3./4 %5!1%/"&/3 6#/72485/480/2/430"2$/&!/"$/&,241.#/&'9&'/&)

.2/ -*3 4,5%/&%&' 6$' 7'%&/5&/&'( 8)/- 9$:;/&'( (8)3:&' (8&) =%/*:+&'0;'0&' " ,'-

>&'?%)0%@ABC)3:&9$:;/&'(+;'0&'6$'

e ⁻

^DE&++&' *,3F+2:G*%:$-&' +)&0&'>&) &)')0&'"H!HHH E&++&'+;'0&'I (! J!

l c ≃ 0, 1µm

^>&)

U = 10kV

^{! K)&} 4>+&'G,'0 &)'&/

e ⁻

DE&++& G*'' -,/:

#*33)&/&'&)'&/L*0'&%)3:&'=:):%-&/K)G&

D

&/5$+0&'I8)&)'4>>! !M-*/0&3%&++%!

!!"#$%&' ():+ !#/&7%&' /"&/4 /!/&/& 6#/7248&/&;/##/ !/"0 <133"/4/& /"&/4 01'&/2"3./&

,.".2)

J&%/*:%&G+&)'&4>+&'G,'0,L-&'E)'G&+

α

^N9/*5%*,5-)&

e ⁻

)''&/:*+>-&/=:):%-):%&

D

^@

F ~ = − e · ~v × B ~

F ~ _x = − e · v · B _z

O3)&:&=G)((&P

¨

x = − e m · v · B z

˙

x = − e

m · v · B z · t

^L)%.+,0(&)%

t ≃ D

v

^{O !QRP}

K*L)%&/0)>%3):52/-&'4>+&'G8)'G&+

α α ≈ x ˙

v ≈ − e m

B z · D

v

^{O !QSP}

12/'.)3,412&,0)+(,/56078,( 95'/,(:;<)&'2/

l

^'15/,23,..,250=2&,2

ds ≃ l · α ≃ − e

m · B _z · D · l

v = − e · ∆φ

m · v

>"#?@A

:(&

∆φ = B _z · D · L

^B,(24,)+.=)),2,0-.1))78()+,2/,2<,(/,2

e ⁻

^C'+2,2#

C,=<'+&124D E(, '<4,.,2F&, 3,..,250=2& .G15& 124,)&H0& 8,(&,0I &0=&7 /,2 /10+.'15,2,2

3,412&,0)+(,/,2

ds

<.,(<& /(, J=+G0,27 ,0+'.&,2#

ds

,2&)K0(+& /'22 ,(2,0 2=&8,2/(4,2 9+'),2L,0)+(,<124710;150,+&,0+'.&124/,0J=+G0,27L=2

∆ϕ = ds

λ · 2π = 2π

|{z} mv

p= ^h _λ

λ · e · ∆ φ = e

~ · ∆ φ.

>"#?!A

-0'4,DMG24&/(,=N,2)(+&.(+'15&0,&,2/,9+'),2L,0)+(,<124

∆ ϕ

⁷¹⁰J=:K,2)'&(=2/,)3,4O 12&,0)+(,/)L=:/10+.'15,2,2

B ~

^=/,0210L=:,(24,)+.=)),2,2-.1))

∆ φ

^{78()+,2/,295'/,2}

12/"'<P

C,'2&8=0&124,05=.4&6<,0Q.,F&0=2,2(2&,05,0=:,&,0,RK,0(:,2&:(& :=/(S7(,0&,: T2&,05,0=:,O

&,05'/,2D

!!"#$%&'()*+, -.$"/0"123124&312512.6131257$1&89'#) !!)()+:;)

E,0T2&,05,0=:,&,05'/,28(0/'15U#M,OV,:K,0'&10,2F6+.<'0'1)4,.,4&12/:(&,(2,0)1K0'.,(O

&,2/,2%+(+&

(P b, T _c = 7.1K)

!"#$%&'(

!"#$%$&'

•

^)*+!,!*-.*

B ~

^{%#/#++!+01$2#"!*}

•

^{) 345+!*"!62#"!*6#1&78}

•

^{) 65#+'!*"!6:1;!/!*}

B ~

^<2!+"!6

(!&)$*+$&,' =!/ $#,*!'>65! 2+166 >6' ?!'0' >$ *./$#++!>'!*"!* 2#"!*>**!/!* !>*,!&/./!*(

=>!

e ⁻

<@'/#5+!*,:*,!6!+ 6'6>*"&!+"&/!>(

-$!./%1+&#' =>! -!/:*"!/'!* A*'!/&!/!*0$16'!/ >* ) 5:*,>,3!>' "!6 !>*,!&/./!*!* 2+166!6

∆ φ

!6':'>,!*">!B4+'>,3!>'"!/. !*6'!5!*"!*B+!>51*,(

∆ϕ = e

~ · ∆ φ

^CD(7EF

C)1;!/"!$+:66'6>50!>,!*G"#66

∆ φ

^{6>5*1/6'1H,>*}I>*5!>'!*"!66.,(2+166J1#*'6

∆φ = ₂ ^h _e

!"#$!%&' ()! *)+*,./)/)& 0&/)!1)!)&2)3)"/)4 2)& *30&+&'4%& 5&++&'6/$'% +7,6%

,'0&8%9/% :&)%&/; %/$%( -&' -,/=+*,6&'&' 5&0,'%&/8=)&-&'

ds

^{3+&)3% -)&} >$=7/&'( &/=*+%&'!

?&)@&/:&'-,'0 -&84*'$')8=&'ABC,+8&8

~p

^(,/ ?&/&=','0-&/-& ?/$0+)&D5&++&'+7'0&&/0)3%

8)=

∆ ϕ

4$//&4%$='&:&)%&/&E''*=B&!

~p =

P ~ + q · A ~

;

^{F !GHI}

~p =

^4*'!ABC,+8

; P ~ =

^B&=!ABC,+8

.

J*B)+)%$'D.,'4%)$'6K/-)&?&:&0,'0)B&+&4%/$B*0'&%)8=&'.&+-L

H = 1 2m

p ~ − q · A ~ ²

+ q · φ

^{F !G"I}

H

&/6K++%-)&4*'$')8=&'M+&)=,'0&'

∂H

∂p i

= ˙ q i ; ∂H

∂q i

= − p ˙ i

^{F !G I}

N,*'%&'B&=*')4OP=/9-)'0&/0+&)=,'0L

i ~ ∂

∂t Ψ =

"

1 2m

~

i ∇ − ~ q · A ~ ²

+ q · φ

#

· Ψ

^{F !GQI}

λ

!"#$%&'!

= h

m · v + q · A _k

^F-&?/$0+)&D5&++&'+7'0&3&) @$/=*'-&'8&)'R$'

A ~

^{I F !GSI}

A _k

^{)8% -)&} >$BC$'&'%& -&8 @&4%$/C$%&'%)*+8 C*/*++&+ (,

~v

^{! 2*B)% +788% 8)= -)& $3)0& ./*0&}

-*=)'0&=&'-3&*'%:$/%&'O-*88',/-&/,B8=+$88&'&OB*0'&%)8=&.+,88,'--*B)%

R A d~s ~

^-)&

#=*8&'3&()&=,'0-&/T+&4%/$'&'3&8%)BB%!

!"!# $%&%'(*+ ,+-.+/ &0 .+/ 1/.23+/45*+

#)"/!56.)67)#(

@$B 5&+%/*,B *,8 3&%/*=%&% &/8=&)'% -)& T/-4,0&+ &+&4%/)8= '&,%/*+! M/,'-L T8 -/)'0&'

+*,6&'-0&+*-&'&' U&)+=&' *,8-&B E++FP$''&':)'-D

e ⁻ , p, α

^{I&)' ,'-8$/0&' 6K/}V*-,'08*,8D 0+&)=!

T)'& A$')8*%)$' -&/ V,6%B$+&4K+& 0&8=)&=% K3&/ P$''&'DW@O X*-)$*4%)R)%7% ,'- >$8B)8=&

P%/*=+,'0!

8)+*,./%&'4 2)& A$'$8C=7/& F"HH D "HHH 4B K3&/ -&/ T/-$3&/Y7=&I +7-% 8)= -*,&/=*6%

C$8)%)R0&0&'-)&T/-$3&/Y7=&*,6LZT8&'%8%&=%&)'>,0&+4$'-&'8*%$/B)%

U

($(

∼ 250kV

^!

2)& .&+-8%7/4& *B ?$-&' 3&%/70% &%:* "HH D QHH @[BO \& '*= 5)%%&/,'0 F3&) M&:)%%&/ 3)8

(, &)')0&' 4@[BI! M&:)%%&/:$+4&' 8)'- R$/ -&/ T'%+*-,'0 )BB&/ 3$-&'8&)%)0 '&0*%)R! 2&/

V*-,'08C/$(&88:)/-K3&/-)& V,6%8%/9B,'0)'-&/M&:)%%&/:$+4&0&%/)&3&','-)8% '$= ')=%

/&8%+$80&4+7/%!

!!"#$%&'()**+ ,%-./$%&'01-2&&%&'"& 324"112-45#62&)

!"#$%&#'()/0$*1$&%,2*.$*&,314&15%'*167.-%$80'9):.*;<%118&$*&1*.*+,&&#'-.'/

0#4*8*''*1=*.-*1"

1,2'#(%-4#- *#+0

!!"#$%&'()*(+ ,%78-$7/'&2192#$)

>.'%)*?@.-%'7.-8%41*$?*,-'&1-A012*<$&01''$.B8*&11*.)%,C-*.)0),*&$*1-*1#)8*,D*

&8 :3E*.*1 7.-<*.1 6

R ∼ 3.000

^<8F G#*,C'$*..*4$*. HI1%80J;F +*,-.&)$314 :1-*.$ '&) @C*.

4.0E*K*&$.:38*"

*#+0-(5$6#')

62µT

^%1-*1/0,*1 +*,-.&)$314'*1<.*)$

31µT

^%8LM3%$0. +*,-.&)$314)0.&D01$%,

!"#$%&#'()N*''314201N%41*$?*,-*.1FO3'8*''314201P*,8)0,$D'93,*18&$ P%,,'01-*"

!"!# $%& '%()*+%,+ -.+,*/0(2*& 3&*0(*

!"#$%&'(")*+!,!")%-!-$)%+-!$

!!"#$%&'()*+, -./!0123"'425423/&4&.3/'&4.3".6474"9&%&'4&)

12 0)+%3

∇ · ~ B ~ = 0

B ~ = µ o ( H ~ + M ~ ) H ~ = B ~

µ _o − M ~

∇ · ~ H ~ = ∇ · ~ B ~ µ o − M ~

!

= − ∇ · ~ M ~

^{4 !556}

7*2#/$8+&9:*''*'*+$0(,/1+&:%/$2%*%):0&+;2%<&/-&'8&)1)'=>?/,'0@$'A:%)@&'B*-,'0&'

8(<!?)&/.+C?&'EB*-,'02-)?%&'

s _m

^,'-B*-,'02-)?%&'

ρ _m

³

∇ · ~ H ~ = ρ m = − ∇ · ~ M ~

^{4 !5F6}

H ~

^:*''<&0&'

∇ × ~ H ~ = ~j = 0

9)%%&+2G:*+*/H$%&'%)*+8&2?/)&8&'<&/-&'3

H ~ = − ∇ ~ Ψ

∆Ψ = − ρ m ;

^I9*0'!#$)22$'EJ+&)?,'0K 4 !5L6

7*9)%&'%2H/)?%-)& B;2,'0 =>/

H ~

^-&9 &+&:%/)2?&' .&+- @$' (<&) :/&)2=;/9)0&' #+*%%&' 9)%

?$9$0&'&/.+C?&'+*-,'02-)?%&!.>/-)&M&/:'>H=,'0@$'

H ~

^,'-

M ~

^9)%

s m

^0)+%3

s _m = M ~ · ~n =

H ~ _a − H ~ _i

· ~n

^{4 !5N6}

→ H ~ i , ~ H a

^2)'--)&.&+-2%C/:&')''&/?*+88(<!*,O&/?*+8-&2#&/9*'&'%9*0'&%2!

G%&%)0:&)%@$'

B ~ n

&/0)8%3

µ ◦ H ~ a = µ ◦

H ~ i + M ~

4 !5P6

1)'2%*8=;/9)0&/Q ?$9$0&' 9*0'&%)2?&/#&/9*'&'%9*0'&%<)/- )9 =$+0&'-&' &)'&/ G$+&'$)-E

GH,+&0&0&'>8&/0&2%&++%!R&)-&S$'A0,/*%)$'&'2$++&'-)&0+&)?&

B(~x) ~

EM&/%&)+,'0(&)0&'!7)&T8E 8)+-,'0&' !"U$8&'V,'%&'(&)0&'-)&/&2,+%)&/&'-&'2%*/:,'%&/2?)&-+)?&'WE.&+-@&/%&)+,'0&'!

7)&2&/M&/0+&)?(&)0%-/*2%)2?-)&T,2<)/:,'0-*@$'Q-*22)'-)&R&/&?','0@$'

H ~

^',/C,O&/&

!!"#$%&'()*+, -!.&,

H ~

^{/0.#$ 1%2} 345!6782"'.&9.825&.&425'&.4: ;&4.&,

H ~

^{/0.#$ 1%8}<=#.&=/

"$3>%#.2"4'.&?@.84 "$.&4"3@.8

B ~

/0.#$B.84."#%&'C". =!.&D !"#$%&')$*$+,$#$-"&$.*$--/0!"#$%&'01'

/*0*1-*'2345,*6*178'%44*1)%10'931

H, ~ ~ B, ~ M

^&4+%,,931:*..34%01*$&')*1#$3;*181-

%8:0.81--*'<1-82$&31'0*'*$=*'48''>*&-*.?*''810>=@!A*.,*&$8109314%01*$&')*.+*,-B

*1*.0&*'3.0:C,$&0 93.0*0%10*1@*.-*1! D&1* >*'31-*.'>*E8*4* F*34*$.&*B%8)=8. G*.4*'B

'810-*.AH'$*.*'*28.9*81--%4&$-*'2345,*6*1 78'%44*1)%10'=@&')*1

H, ~ ~ B, ~ M

^B,&*:*.$

-&*I&10'58,*4&$0*'),3''*1*4:*..34%01*$&')*4J*.1!

78.G*.*&1:%)810'*&-*.J*.1'*).'),%12K

R = r a + r i

2 ; r a − r i

R ≪ 1;

H o ≃ N · I

2πR

^{L !M"N}

!!"#$%&'()*+, -"&'./0&'/12/30"/ 2"340"2506 %&$7/.%&$5089%#/)

2&3')%)$'4$'

H s

⁵6%/&,7&+-8

H ~ = H ~ ◦ + H ~ s ; H ~ ◦

^)9%-*9.&+-(,&)'&/0&0&:&'&'

~j(~x)

^;<&/%&)+,'0)=<*>,,= ? !@"A .B/-&'$:&'-&3')&/%&'C)'0>&/'9&)-)&#&/=&*:)+)%D%E$=$0&'5-!E!-*96%/&,7&+-4&/9EG)';

-&%8

H ~ = H ~ _◦

^{? !@ A}

H&).+,99D'-&/,'0&')=C)'0I,&/9E')%%

A

^0)+%*+9J'-,>%)$'99K*'','0K/$L)'-,'08

U _i = ˙ φ = ˙ B · A

^{? !@1A}

J'%&0/*%)$'B:&/-)&M&)%&/0):%8

Z t

◦

U _i dt = B(t) · A;

^=)%

B = 0

^7B/

t = 0

^{? !@NA}

.B/7&//$=*0'&%)9E&6%$O&0)+%8

B(t ~ 0 ) = µ ^∗ · H(t ~ 0 )

^{)9%-*:&)4$'-&/<}$/0&9E)E%&

H(t), B(t)

^7B/

t < t 0

^{*:ED'0)0 ? !@PA}

!!"#$%&'()*:, ;%2<=83/0/8/>/0?@#3/& >1&731A/&)

Q)%%&+9 -&/ 6K,+&'9$'-& ?9! 6>)((& $:&'A >*'' -)& RS9%&/&9&>,/4& &)'&9 C)'0>&/'9 :&9%)==%

G&/-&'!

H&/&E','0 -&/ =*0'&%)9E&' .&+-&'&/0)&8 2)& *' -&/ #/)=D/9K,+& )'-,()&/%& 6K*'','0

:&).+,99D'-&/,'0)9%

U i = N · φ ˙ = N · A · B. ˙

^{? !@@A}

01%(2/+*+1.'(%.'+&345+6/&*%+7+'(%31583/,+-.91.+/315

dW

dt = I · U _i = IN A B ˙ = IN

2πR A2πR

| {z }

!"#$%&

B ˙ = H · V · B ˙ 1

V · dW

dt = H · dB dt 1

V · dW

| {z }

⇓

= H dB

01+/5'+.'*%+91.+/315

w 1,2 = Z B 2

B 1

HdB

: ";<=

!!"#$%&'()*+, -."$./ 01'&.2"3"./%&'1%4'.51&$2. /!."2)

w 1 , 2 = W 1,2

V =

>1.+/315.+/01+/5'+.'*%+6+'?@6+/5&15AB1

B 1

^&34

B 2

^{: ";C=}

!!"#$%&'()*6, 7./1&391%#"9%&' $./:;;1'&.2"3"./%&'3<./#%32.=/> ?@A#%3)

.2/ 3&-)&' 4)% +)'&*/&4 5,6*44&'7*'0

B( ~ H) ~

^{0)+% 8(0+! -&/} 4*0'&%)67&' :'&/0)&-)7;

%&<

.*++"<

.*++6=6$%/$>)&0)+%<

B ~ = µ · H ~ w =

Z B 0

HdB = 1 µ

Z B 0

BdB = B ² 2µ = µ

2 H ²

^{? !@AB}

.*++ <

C&)')7%)6$%$>&'3&-)&'D8(E!*++0&4&)'<

w = 1

2 B ~ · H ~

^{? !FGB}

H'E&'-,'0(,/C&/&7','0-&/4*0'&%)67&'I/*J%(E)67&'H'K&/,'-

U

;JL/4)0&4M$7<

!!"#$%&'()*+, -%. &/"01%&'23.456 /7"210&9:5;.<"'0< =>1%&$ &30.)

N&O')%)$'-&64*0'&%)67&'P)-&/6%*'-6<

R m = H Hds

φ

^{!# ? !F"B}

I/*J%*,JH'K&/<

F · ds = dW

$%&'(

= w · 2A · ds F = 2w · A = B ²

µ _◦ · A F = φ ²

!#

µ ◦ · A ;

^{? !F B}

φ

^{!# 7Q'0%6%*/KR$'}

s

^{*8ST0+!-*(,-&'}'Q76%&'H867')%%S

!"#$%&')$*+!##,#" ,#- .!"#$%&')$/ 0&-$/'%!#-

01&-23'+45'(*+1(%&%'(*+16+-+7%/'(*+181.9&31+%'(*+1:/+'(+1;

<-+7%/'(*+/$%/29 =&31+%'(*+/,-8((

I = R ~j · d~a φ = R B d~a ~ R

A ~j d~a = 0 H

A B d~a ~ = 0

<-+7%/'(*+$>&11813 =&31+%'(*+$>&11813

U = R E d~s ~ U _m = R H d~s ~ H

s E d~s ~ = 0; H

s H d~s ~ = N · I

<-+7%/'(*+/?'.+/(%&1. =&31+%'(*+/?'.+/(%&1.

R = ^U _I R m = ^U _φ ^m

@'+ :'/**2A(*+1 B+3+-1 7C11+1 DE+/%/&3+1 5+/.+1 F 9'% .+/ G8(&%4E+.'13813 E43-" .+/

=&(*+1/+3+-

H

s Hds ~ = N · I

^H

I+'(>'+-;JE+1E+(>/2*+1+/K2/8(9'%B'137+/1

!!"#$%&'()(*+ ,-.0#"121-3 4"&'5-3&6"17%81.9:#1)

•

^{1!22 345)#$ 6,7%'+!2%4}

φ = B · A = µ _r µ _◦ H · A

= H · l _T

| {z }

U m

· µ r µ ◦ A l T

⇒

=&31+%'(*+/?'.+/(%&1.

R _m = l T

µ r µ ◦ A

^{L "MNO}

@&483+*C/%+'1+9&31+%'(*+P+'%QR*'37+'%

σ m = µ r · µ ◦

^{L "MSO}

•

!""#$ %&'()*+' ,&- .0,!"*,123-.4!"-

l L

R m,L = l _T − l _L µ r µ ◦ A + l _L

µ ◦ A R m,L = l T + l L (µ r − 1)

µ r µ ◦ A

^{2 !134}

5&/%&)+,'0-&/.&+-&/6

∇ · ~ B ~ = 0;

7%&%)08&)%9$'

B _n

^*'-&':/&'(;<>&'

→ B ϕ = const;

^?@/0&A*B%&'CB?*'0 2 !1D4 EF><'0)8&)%9$'

| B _ϕ |

^9$BG,?%AH*+%

l _L

⁶

B ϕ (l L ) = B ϕ (l L = 0) · R m

R _m,L

^{2 !114}

I)%

B = µ _◦ µ _r H

^)BJ&/'9$+,B&'?$+0%K-*AA&'%AH/&>&'-

µ _r

^-*ALM.&+-&/>&F+)>')&-/)0&/*+A )BG,?%AH*+%)A%!

H

s Hd~s ~ = N I = const B ϕ = µ ◦ · H L = µ ◦ µ r H

^!"#

H L

H

!"#

= µ r µ ◦

µ ◦

= µ _r ;

^{2 !1N4}

564*+&,*'-6I&AA,'0-&ALM.&+-&A*'9&/A>)&-&'&'O$>MJ$'P0,/*%)$'&'