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1 2008 II

2 i I

3 ii B H E D B H

4 1 0 I II 0.1 I (source) source 1 D = ε 0E + P (1) D P div E = ρ div D ε = ρ (2) 0 1

5 N N N S N S Wb 2 m 1 [Wb] m 2 [Wb] r[m] F = m 1m 2 4πµr 2 F = m 1m 2 4πµ 0 r 2 (3) µ µ 0 MKSA µ 0 = 4π 10 7 MKSA [A] 3 E[N/C] H[N/Wb] [Wb 2 /N m 2 ] [Wb] [N m/a] [N/A 2 ] SI [m][kg][s][a] [N/A 2 ] [H] [H/m] H q F = q E F = q 1 q 2 4πε 0 r 2 [C] [N/C] [V/m] m F = m H F = m 1m 2 4πµ 0 r 2 [Wb] [N/Wb] [A/m] N N N S Oersted Ampere 5 2 A A/m F = m H Wb 3 4π ON/OFF 5 cgs H cgs

6 N S N S H H H H 6 H (1) (2) (3) (3) 7 (3) (1)(2) (3) (3) 6 7

7 4 0 ( ) 8 N S

8

9 N N (E-H ) N S (E-B ) S N N S N S N S E H E-H E-B E-B E D H B 13 B 14 B E, H 14 H E-H E-B B 15 B B B magnetic induction

10 B I F = I B (4) B x x + d x d F = Id x B (5) d F d H B B = µ 0 H B N/A m B T Wb Wb/m 2 E B a b a b a b a, b θ a b a b = a b sin θ a b sin θ a b E I B I d F B B I (4) B r 2 (I 1 I 2 ) µ 0I 1 I 2 2πr

11 8 0 N S N S I (1) E V (2) V E (3) E (4) (5) Q E = Q 4πε 0 r 2 e r 0-2 I (1) div,rot,grad (2) grad rot 0 rot div 0 (3) e r r + 1 r e θ θ + 1 r sin θ e φ φ (4) = 1 ( r 2 r 2 ) ( 1 + r r r 2 sin θ ) sin θ θ θ r 2 sin θ φ 2 (5) V = ρ ε 0 ρ

12 I[A] r I 2πr φ e φ H = I 2πr e φ (1.1) e φ z B = µ 0I 2πr e φ (1.2) I r 1 1 rot E 0 rot E 0 rot E 0

13 div B = 0 3 H rot H = 0 H = I 2πr e φ rot A A rot A rot H r m[wb] m I 2πr m I 2πr = mi (1.3) 2πr r A B m I m θ r θ = (1.4) 2πr 2π B C D A C D m I (r + r) θ = m θ 2π(r + r) 2π (1.5) r + r 0 mi 0 R A r θ D r r N B N N C 2 3 div H = 0 div B = 0 div B = 0

14 I[A] m[wb] mi[j] [Wb A]=[J] [C V]=[J] 4 ds m rot H ds 5 j mrot H ds = m j ds (1.6) ( ) rot H j ds = 0 d S 0 rot H j = 0 mds 6 rot H = j (1.7) rot B = µ 0 j rot H = j div rot div B = 0 div D = ρ rot E = 0 div B = 0 rot H = j div rot 4 5 m H d S rot (m H) d S m rot 6 md S

15 12 1 div D = ρ rot E = 0 rot H = j div B = 0 rot E = 0 E = grad V rot 0 grad div B = E V E = grad V H V m H = grad V m rot H 0 8 V=4 V=3 V=5 V=0 V=2 V= div 0 rot div B = 0 B B = rot A A 8 cut

16 EFGH 0 F G H E E F G H EF GH ABCD C D 0 A B AB L H HL ABCD n I nl nli HL = nli H = ni (1.8) D C A B H G E F B C D C A B d z z = d z = d j x x, y z x yz z z z z > 0 z < 0 div H = 0 z y z > d rot H = 0 H y z = d rot H = j x H z //// y x z z = -d H y z = j (1.9) y j H y = jz jd d < z H y = jz d < z < d jd z < d (1.10)

17 14 1 rot H 0 rot H = 0 z y H = I 2πr e φ x = 0, y = 0 rot H = I N z 1-3 V m = I 2π φ z grad (ρ, φ, z) H = I 2πρ e φ = e ρ ρ + e z z + e 1 φ ρ φ R r H = grad V m V m 1-4 d v ±σ (1) (2) 1 2ε 0 σ rot H = j S d x H = ( S S S S d S j (1.11) S d x ) S σ σ v v

18 E div D = ρ rot H = j D, H ρ, j D, H ρ E Q x x E Q = 4πε 0 x x 2 e x x x ρ d 3 x ρ( x )d 3 x E( x) = div E = ρ ε d 3 x ρ( x ) 4πε 0 x x 2 e x x (2.1) (2.1) div div E( x) = ρ( x) = ( d 3 x ρ( x ) 1 4πε 0 x x 2 e x x ( ) d 3 x ρ( x ) 1 4π x x 2 e x x ) div E = ρ ε 0 ε 0 (2.2) 1 ε 0 2 (2.1) π x x 2 e x x div

19 16 2 ( ) ρ( x) x x 1 4π x x 2 e x x x x x x f( x) d 3 x f( x )δ( x x ) = f( x) (2.3) δ( x x ) ( ) δ( x x ) = 1 4π x x 2 e x x (2.4) x x 0 x = x d 3 x δ( x x ) = 1 (2.5) 1 1 4π x x 2 Q x ρ Q ρ 0 Q B div E = ρ (2.1) ε 0 B( x) ( ) = d 3 x j( x ) x x (2.6) I j( x ) x x x j( x ) B( x) j x x x x j ( x x ) j ( x x ) j x x j x x

20 B( x) = K d 3 x j( x ) ( x x ) x x n (2.7) K n n = 3 n = 2 x x n = 3 B( x) = K d 3 x j( x ) ( x x ) x x 3 = K E( x) = d 3 x j( x ) e x x x x 2 (2.8) d 3 x ρ( x ) 4πε 0 x x 2 e x x ρ j x x B = µ 0I 2πr e φ K z j z dx dy dz j z 4 dx dy j z I z x = y = 0 0 x = y = 0 x = z e z dz x = r e r + z e z x x = r e r + (z z ) e z (2.9) I e z e z I ( x x ) = I( e z r e r ) = ri e φ B(r) = K dz Ir e (r 2 + (z z ) 2 ) 3 φ (2.10) 2 z z z = z = z z = 0 B(r) = K dz Ir e (r 2 + (z ) 2 ) 3 φ (2.11) 2 Z 4 Z d 3 x dx Z Z dy dz

21 18 2 z = r tan θ 5 θ π 2 π 2 I B(r) = KI r π 2 π 2 dθ cos 2 θ 1 (1 + tan 2 θ) 3 2 e φ = 2KI r e φ (2.12) B = µ 0I 2πr e φ K µ 0 4π 6 Biot-Savart (Biot) (Savart) 1820 r z z j( x) x B( x) B( x) = µ 0 4π d 3 x j( x ) ( x x ) x x 3 = µ 0 4π d 3 x j( x ) e x x x x 2 (2.13) 7 div j = 0 div j = ρ ρ t 5 z = 0 z z = r tan θ 6 µ 0 4π

22 µ 0 E Q = 4πε 0 r 2 e r ε 0 B = µ 0I d x ( x x ) 4π x x 3 µ 0 SI B L I F = BIL F = m 1m 2 4πµ 0 r 2 µ rot H = j rot rot B = d 3 x µ 0 j( x ) e x x 4π x x 2 (2.14) A ( B C) = B( A C) C( A B) A ( B C) = B( A C) ( B A) C ( A }{{} j ( x ) }{{} B ( )) e x x } 4π x x {{ 2 } C ( = j( x ) A } {{ } }{{} B ( e x x )) 4π x x } {{ 2 } C ( j( x ) } {{ } B A }{{} ) ( ) e x x 4π x x } {{ 2 } C ( ) x 8 j( x e x x ) x x x 2 x A ( ) ( ) e x x C 4π x x 2 A ( ) B ( ) j( x ) B C (2.15) B B C ( B C) = B ( ) ( ) ( ) ( ) C + C B B C C B (2.16) rot B = µ 0 d 3 x ( j( x ) ( ) ( )) e x x 4π x x 2 j( x ) e x x 4π x x 2 (2.17) 0 e x x 4π x x 2 x x x x ( ) ( ) µ 0 d 3 x j( x ) e x x 4π x x 2 = µ 0 d 3 x j( x ) e x x 4π x x 2 (2.18) x ( ) µ 0 d 3 x j( x ) e x x 4π x x 2 = µ 0 «8 x = (x, y, z) = x, y, z ( ) d 3 x j( x ) e x x 4π x x 2 (2.19)

23 20 2 j 0 j( x ) = div j( x ) = 0 divergence 0 (2.17) 0 9 ( ) 1 4π x x 2 e x x rot B( x) = µ 0 d 3 x j( x )δ 3 ( x x ) = µ 0 j( x) (2.20) B = µ 0 H rot H = j rot H = j I I I 0 x dx dy dz j x e x ( ) (2.21) dy dzj x I y, z n dx I e x ( ) (2.22) x dx x dx y z dy, dz (Idx e x + Idy e y + Idz e z ) ( ) (2.23) I d x dx dy dz j ( ) I d 3 x j d x ( ) (2.24) d x (dx, dy, dz) d x = dx e x + dy e y + z e z FAQ x 9 div j = 0 div j + ρ t = 0

24 j 0 j d x A (2.25) ( ) d x A ( ) d x A ( ) d x A x y z = = = (dya z dza y ), (dza x dxa z ), (dxa y dya x ) (2.26) I x B( x) B( x) = µ 0I 4π d x ( x x ) x x 3 (2.27) I x m x B = m x x 4π x x 3 = m 4π x x 2 e x x (2.28) I d x F = Id x B N F = mi d x ( x x) 4π x x 3 (2.29) F = mi d x ( x x) 4π x x 3 = mi d x ( x x ) 4π x x 3 (2.30) m H 10

25 R I x d x x x 360 z x = z e z x x Id x x x x x Id x x z = 0 +z z 0 z 0 z z x x R φ 0 2π x d x Rdφ d x x x d x ( x x ) d x Rdφ x x R2 + z 2 µ 0 IRdφ 4π(R 2 + z 2 ) φ z y (2.31) R z 2π 0 R µ 0 IRdφ R2 + z 2 4π(R 2 + z 2 ) µ 0 IR 2 µ 0 IR 2 dφ = 4π (z 2 + R 2 ) (z 2 + R 2 ) 3 2 (2.32) (2.33) z 3 z = 0 z = 0 B( 0) = µ 0I (2.34) 2R R π

26 d x = Rdφ e φ x x = z e z R e r e r, e φ x 11 Id x ( x x ) = IRdφ e φ (z e z R e r ) = IR(z e r + R e z )dφ (2.35) 12 e r, e φ, e z e r e φ = e z, e φ e z = e r, e z e r = e φ (2.36) B(z e z ) = µ 0 4π 2π 0 dφ IR(z e r + R e z ) (z 2 + R 2 ) 3 2 φ (2.37) B(z e z ) = µ 0 4π IRz (z 2 + R 2 ) 3 2 2π 0 dφ e r + µ 0 4π IR 2 e z (z 2 + R 2 ) 3 2 2π 0 dφ (2.38) e z e r 2π 0 dφ e r = 0 B(z e z ) = µ 0 2 IR 2 e z (z 2 + R 2 ) 3 2 (2.39) z z x x y 13 z x = x e x + z e z (2.40) x x φ R z x = R cos φ e x + R sin φ e y = R e ρ (2.41) e ρ z ρ d x d x = Rdφ ( sin φ e x + cos φ e y ) = Rdφ e φ (2.42) e φ 11 e e 12 0 d x x = 0 d x x 13 y

27 24 2 x x x x = z e z + x e x R(cos φ e x + sin φ e y ) (2.43) φ x x φ x x = z 2 + R 2 + x 2 2Rx cos φ (2.44) z xy z z B( x) = µ 0I 4π Rdφ eφ (z e z + x e x R e ρ ) (z 2 + R 2 + x 2 2Rx cos φ) 3 2 (2.45) e φ e z = e ρ, e φ e ρ = e z (2.46) e φ e x e φ = sin φ e x + cos φ e y e x e x = 0, e y e x = e z 14 B( x) = µ 0I 4π e φ e x = cos φ e z (2.47) Rdφ (z eρ x cos φ e z + R e z ) (z 2 + R 2 + x 2 2Rx cos φ) 3 2 ρ z φ R = x cos φ z 0 x x xy R z, x R 1 (z 2 + R 2 + x 2 2Rx cos φ) 3 2 B( x) = µ 0I 4π = = φ (2.48) ( ) 1 (z 2 + R 2 + x 2 2Rx cos φ) 3 2 R=0 + R 1 R (z 2 + R 2 + x 2 2Rx cos φ) 3 R= xR cos φ + + (z 2 + x 2 ) 3 2 (z 2 + x 2 ) 5 2 (2.49) ( ) 1 3xR cos φ Rdφ (z e ρ x cos φ e z + R e z ) + + (z 2 + x 2 ) 3 2 (z 2 + x 2 ) 5 2 (2.50) 14

28 R µ 0 I 4π 1 Rdφ (z e ρ x cos φ e z ) (z 2 + x 2 ) 3 2 (2.51) φ φ e ρ cos φ 0 2π dφ e ρ dφ cos φ 0 R R 2 µ 0 I 1 Rdφ R e z + µ ( ) 0I 3xR cos φ Rdφ (z e 4π (z 2 + x 2 ) 3 ρ x cos φ e z ) 2 4π (z 2 + x 2 ) 5 2 µ 0 IR 2 e z = dφ + 3µ 0IxR 2 ( ) (2.52) z dφ cos φ e 4π (z 2 + x 2 ) 3 2 4π (z 2 + x 2 ) 5 ρ x e z dφ cos 2 φ 2 dφ = 2π, dφ cos φ e φ = π e x, dφ cos 2 φ = π 15 µ 0 IR 2 e z 2 (z 2 + x 2 ) µ 0IxR 2 4 (z 2 + x 2 ) 5 2 (z e x x e z ) = µ 0IR 2 e z 4 (z 2 + x 2 ) 5 2 ( 3xz ex + (2z 2 x 2 ) e z ) (2.53) z p ( ) p z 1 E = 3(x e x + y e y + z e z ) e 4πε 0 (x 2 + y 2 + z 2 ) 5 z 2 (x 2 + y 2 + z 2 ) 3 2 p ( 3xz ex + (2z 2 x 2 ) (2.54) ) e 4πε 0 (x 2 + z 2 ) 5 z 2 y = 0 (2.53) (2.54) p IπR 2 q q l q l q l E E (q l) Z 15 e ρ = cos φ e x + sin φ e y dφ cos φ sin φ = 0

29 26 2 m m l 1 µ 0 m l 1 µ 0 E B( H ) p = I S (2.55) S IL 2 = ml µ 0 (2.56) z = 0 z = l z z l z l B( x) = µ 0I 4π Rdφ (z eρ x cos φ e z + R e z ) (z 2 + R 2 + x 2 2Rx cos φ) µ 0I 4π Rdφ ((z l) eρ x cos φ e z + R e z ) ((z l) 2 + R 2 + x 2 2Rx cos φ) 3 2 (2.57) z B( x) = µ 0I 4π n= Rdφ ((z nl) eρ x cos φ e z + R e z ) ((z nl) 2 + R 2 + x 2 2Rx cos φ) 3 2 (2.58)

30 n= l n= l dz (2.59) l B( x) = µ 0nI 4π Rdφ (z eρ x cos φ e z + R e z ) dz (z 2 + R 2 + x 2 2Rx cos φ) 3 2 (2.60) z z 0 z e ρ z B( x) = µ 0nI 4π dz 2π 0 R ( x cos φ + R) e z dφ (z 2 + R 2 + x 2 2Rx cos φ) 3 2 z = R 2 + x 2 2Rx cos φ tan θ dz = R 2 + x 2 2Rx cos φ dθ cos 2 θ B( x) = µ 0nI 4π = µ 0nI 4π = µ 0nI 4π π 2 π 2 dz dθ R 2 + x 2 2Rx cos φ 1 cos 2 θ π 2 π 2 dθ cos θ 2π 0 2π 0 2π 0 R ( x cos φ + R) e z dφ (z 2 + R 2 + x 2 2Rx cos φ) 3 2 R ( x cos φ + R) e z dφ( (R 2 + x 2 2Rx cos φ) (1 + tan 2 θ } {{ } = 1 cos 2 θ dφ R ( x cos φ + R) e z (R 2 + x 2 2Rx cos φ) ) 3 2 ) (2.61) (2.62) θ π 2 dθ cos θ = 2 π 2 φ dφ R ( x cos φ + R) e z (R 2 + x 2 R x 2Rx cos φ) dφ AB Rdφ P AC AP AP = R2 + x 2 2Rx cos φ AD = x cos φ R E D ABC APD AC = AB x cos φ R R2 + x 2 2Rx cos φ (2.63) B A C O P AC AP = Rdφ x cos φ R R 2 + x 2 2Rx cos φ (2.64) AB dφ x > R 0 2π 0 0 x < R 2π µ 0nI 4π 2 2π = µ 0nI

31 r I r 2-2 2a I 2-3 a 2b a, b x x 2 (x 3 ) 2-4 y = 1 4a x2 I x = 0, y = a 2a θ r = r dθ 1 + cos θ y dy 2b x a z y=a θ dx x 2-5 N z 2R z N + 1 N 2R

32 I 1, I 2 r I 1 I 2 I 2 2πr µ 0I 2 2πr l F = I 1 µ 0I 2 2πr l = µ 0I 1 I 2 l 2πr (3.1) SI µ 0 = 4π 10 7 F = I 1I 2 l (3.2) r [A] j 1 V 1 j 2 V 2 j 1 x 2 B 1 ( x 2 ) = µ 0 j d 3 1 ( x 1 ) ( x 2 x 1 ) x 1 4π V 1 x 2 x 1 3 = µ 0 j d 3 1 ( x 1 ) e x x 1 x 2 1 4π V 1 x 2 x 1 2 (3.3) x 2 j 2 ( x 2 ) F j1 j 2 = d 3 x 2 j 2 ( x 2 ) B 1 ( x 2 ) V 2 ( ) = µ j 2 ( x 2 ) j 1 ( x 1 ) e (3.4) 0 d 3 x 1 d 3 x1 x 2 x 2 4π V 1 V 2 x 2 x A 3000 r 1cm F cgs (dyne) 2

33 30 3 A ( B C) = B( A C) C( A B) j 2 ( x 2 ) ( j 1 ( x 1 ) e x1 x } {{ } } {{ } } {{ } 2 A B C ) ( = j 1 ( x 1 ) j 2 ( x 2 ) } {{ } } {{ } B A j 1 ( x 1 )d 3 x 1 ( ) j 1 ( x 1 ) j 2 ( x 2 ) e x1 x 2 ( ) j 2 ( x 2 ) j 1 ( x 1 ) e x2 x 1 e x1 x 2 } {{ } C ) e x1 x } {{ } 2 C ( j 1 ( x 1 ) } {{ } B 1 ) j 2 ( x 2 ) } {{ } A µ 0 d 3 x 1 d 3 x 2 j 1 ( x 1 ) j 2 ( x 2 ) e x 1 x 2 4π V 1 V 2 x 2 x 1 2 = µ ( ) 0 d 3 x 1 d 3 x 2 j 1 ( x 1 ) j 2 ( x 2 ) 4π 1 2 V 1 V 2 x 2 x 1 (3.5) (3.6) e x1 x 2 x 2 x 1 2 = 2 ( ) 1 x 2 x 1 (3.7) 2 2 x x µ 0 d 3 x 1 d 3 x 2 j 1 ( x 1 )j 2 ( x 2 ) ( ) 1 = µ 0 d 3 x 1 d 3 x 2 j 1 ( x 1 ) j ( ) 2( x 2 ) 1 4π V 1 V 2 x 2 x 2 x 1 4π V 1 V 2 x 2 x 2 x 1 (3.8) y z ( ) µ 0 d 3 x 1 d 3 x 2 j 1 ( x 1 ) 4π 1 j 2 ( x 2 ) V 1 V 2 x 2 x 1 (3.9) V 2 j 2 0 x 2 x 1 3 div j = j = div j = 0 0 µ 0 d 3 x 1 d 3 x 2 j 1 ( x 1 ) j 2 ( x 2 ) e x 1 x 2 4π V 1 V 2 x 2 x 1 2 (3.10) 2 E = Q 4πε 0 r 2 e r V = Q 4πε 0 r E = V 3 div j 0

34 ρ 1 V 1 ρ 2 V 2 1 d 3 x 1 d 3 x 2 ρ 1 ( x 1 )ρ 2 ( x 2 ) e x 1 x 2 4πε 0 V 1 V 2 x 2 x 1 2 (3.11) 1 ε 0 ρ 1 ( x 1 )ρ 2 ( x 2 ) µ 0 j 1 ( x 1 ) j 2 ( x 2 ) (3.12) jd 3 x V Id x L µ 0I 1 I 2 4π L 1 e x d x 1 d x 1 x 2 2 L 2 x 2 x 1 2 (3.13) j = ρ v 1 d 3 x 1 d 3 x 2 ρ 1 ( x 1 )ρ 2 ( x 2 ) e x 1 x 2 4πε 0 V 1 V 2 x 2 x 1 2 µ 0 d 3 x 1 d 3 e x1 x x 2 (ρ 1 ( x 1 ) v 1 ( x 1 )) (ρ 2 ( x 2 ) v 2 ( x 2 )) 2 4π V 1 V 2 x 2 x = d 3 x 1 d 3 e x1 x x 2 ρ 1 ( x 1 )ρ 2 ( x 2 ) (1 ε 0 µ 0 v 1 ( x 1 ) v 2 ( x 2 )) 2 4πε 0 V 1 V 2 x 2 x 1 2 ε 0 µ 0 1 ε0 µ 0 (3.14) 3.3 Id x B d F = Id x B (3.15) ( ) dv ρ v 4 dv ρdv 4 v

35 32 3 q ρdv q d x ds 5 dv ds d x Id x Id x = ( j ds)d x = j(d S d x) = ρ vdv (3.16) j d x ( j ds)d x = j(d S d x) 6 j = ρ v F = ρdv v B F ρdv q = q v B (3.17) 0 q v B q E ( ) F = q E + v B (3.18) 7 q v B 8 () () r m v2 r = qvb (3.19) ω = v r = qb m (3.20) ω ( ) mv = qbr (3.21) 9 r qvb v 5 6 j ds θ ( j ds)d x j(d S d x) j ds d x cos θ 7 q v B 8 9

36 T = 2πr v = 2πm qb (3.22) r 10 A B

37 x z z y x ( x ) z v evb y y y y V y V d y x x y evb = e V d V = vbd (3.23) vbd 12 x 13 v v I I = envs S n n F = q v B y x Hall (hole) 13

38 V ( E = grad V ) V m H = grad V m rot H E rot E = 0 rot 0 grad E = grad V V E = grad V V E rot Hrot B 0 V m div B = 0 div 0 rot A V rot A + grad V

39 36 3 B = rot A (3.24) A A U = j A (3.25) A j A > 0 rot z z rot A rot A rot

40 field field rot E = grad V V B = rot A A rot j A A d 3 x j A = I d x A N S

41 38 3 rot j A A j qv j j A 1 2 j 1 A 1 j 2 j 2 A 1 j 2 A 2 A 1 j 1 A 2 (overcounting) E = grad V V V + V 0 V 0 B = rot A rot 0 A B rot 0 grad Λ Λ grad grad rot 0 A A + grad Λ (3.26) 16 j A A d 3 x j A d 3 x j A d 3 x j grad Λ = d 3 x j A + d 3 xdiv jλ + ( ) (3.27) 16

42 div rot B = µ 0 j B = rot A ( rot rot A ) = µ 0 j (3.28) rot ( rot V ) = grad ( div V ) V (3.29) grad ( div A ) A = µ 0 j (3.30) div A = 0 A Λ div ( A + grad Λ) = 0 div (grad Λ) = Λ Λ = div A Λ V = ρ ε 0 ε 0 div A A = µ 0 j (3.31) V = ρ ε 0 (3.32) V ( x) = 1 4πε 0 d 3 x ρ( x ) x x A( x) = µ 0 4π d 3 x j( x ) x x (3.33) (3.34) «18 1 4π x x = δ 3 ( x x ) (3.33) (3.32) (3.34) (3.31) 19

43 40 3 ρ I r r I ρ V = 1 2πε 0 log r A = µ 0 2π log r e z (3.35) 20 z z V E A B B = rot A = µ 0 2πε 0 E = grad V = e r (log r) = 2πε 0 r 2πε 0 r e r (3.36) ( e r r ) (log r e z ) = µ 0 1 2π r e r e } {{ z } = e φ = µ 0 1 2π r e φ (3.37) x E E y B B q (1) x, y, z v x, v y, v z (2) v x d2 v x dt 2 (3) v z v z + (4) 20 z

44 m q F = mω 2 r e r r F = mω 2 (x e x + y e y + z e z ) (3.38) m d2 x dt 2 = mω2 x, m d2 y dt 2 ω = mω2 y, m d2 z dt 2 = mω2 z (3.39) z B B = B e z q v B (1) (2) z x, y X = x + iy x, y (3) X = e iωt Ω (4) 3-3 z B v v m q z v xy v (1) z z B z B 0 z z B 0 + B (div B = 0 ) B (2) z z v (3) B v xy z z v (4) z 1 2 m ( (v ) 2 + (v ) 2) (5) z

45 B µ µ B N +m -m I I S -m S z z θ V m = 1 Bz µ 0 mv m (z), mv m (z + L cos θ) +m A = (0, Bx, 0) I A a S -m N x z y

46 E = 1 ( ) D P (4.1) ε 0 H = 1 ( ) B Pm (4.2) µ 0 P m 2 (diamagnetism) 3 (paramagnetism) 4 (ferromagnetism) 5 N S S N diamagnetism dia-diameter 4 paramagnetism para parallel para 5 ferro-

47 44 4 I S IS +m m l ml µ 0 IS [Am 2 ] [A/m] ( ) M M = χ H = χ µ 0 B (4.3) χ 6 magnetic susceptibility 4.2 πr 2 q v v 2πr qv 2πr p m = qv 2πr πr2 = qvr 2 L = mvr p m = 7 (4.4) q 2m L q 2m 8

48 mrω 2 = e2 ± erωb (4.5) 4πε 0 r2 ± ω mr (ω 2 e ) m ωb e2 4πε 0 r 2 = 0 ) 2 + r e2 B 2 4m + e2 4πε 0 r 2 = 0 ( ω eb 2m ( mr ω eb 2m ) 2 = e2 B 2 4m 2 r + e 2 4πε 0 mr 3 ω ω ± eb 2m eb m r ( (4.6)

49 46 4 ) eb m χ 10 6 a χ = 1 a ()

50 e kt E 14

51 a a b 0 0 c b a d e

52 4.5. B H 49 (1) (2) (3) 4.5 B H E D B H E D E D div P ρ ρ P = div P div E = 1 (ρ ε div ) P 0 x ) div (ε 0E + P = ρ (4.7) ε 0E + P = D div D = ρ 1 P ε0 D ε 0 1 div P ε 0

53 50 4 P E D D = εe E D E 1 P E ε 0 ε 0 E D D 15 D B H H B B H B B 1 µ 0 µ 0 B = µ 0 H j ( ) B rot = j + j M (4.8) j j M µ 0 ρ P P j M M M 4.1 M 4.1 x y z M x M x z j z y M x j z = y M x P P t 17 H B B µ 0 µ 0 H

54 4.5. B H : dy z dx dz y x dxdy dxdy M x y j z M y x j z z M x y M y x x M y y M x = j z (4.9) j x, j y 18 rot M = j M (4.10) z ( ) B rot = j µ + j M 0 }{{} ( ) =rot M (4.11) B rot M µ = j 0 x y 18 x y, y z, z x xyz

55 52 4 H H B µ 0 M (4.12) rot H = j B rot ( ) B = j + j M, div B = 0 (4.13) µ 0 H rot H = j, div H = ρ M (ρ M = div M) (4.14) ρ M ρ P = div P rot H = j, div B = 0 (4.15) ( j ) E B D H E(Electric Field) B(Magnetic Induction) D(Electric Displacement) H(Magnetic Field) B H M = χ H H = B χh µ 0 (4.16) (1 + χ)µ 0H = B (1 + χ)µ 0 µ χ µ µ χ µ r B = µ H = µ r µ 0 H (4.17) χ 1 1 H M H = 0 M 0 B H B B H B Magnetic Flux Density D Electric Flux Density 20 Electric Displacement() Magnetic Induction()

56 M H B x µ xx µ xy µ xz H x B y = µ yx µ yy µ yz H y (4.18) B z µ zx µ zy µ zz H z H B 4.6 rot M 0 rot M 0 div B = 0 H rot H = 0 B H µ 0 NNNNNNNNNN SSSSSSSSSSS M 1 µ 0 B H = M B 1 µ 0 H M H B H B

57 div D = ρ, rot E = 0, div B = 0, rot H = j ρ, j D, B div 0 E, H rot 0 div 0 div 0 0 B rot 0 0 H D, B E, H div rot µ n I 4-2 µ 1 µ 2 (1) H B (2) (3) (4)

58 µ H B d θ θ = 0 θ = π 2

59

60 57 5 div D = ρ F = Q E V V = 1 ε 0 ρ E = V rot H = j F = Q v B A A = µ0 j B = A = Il B flux 3 Φ B Φ = B ds (5.1) S S div B = 0 ()

61 58 5 θ V = dφ dt (5.2) V Φ Φ Φ 4 Φ Φ Φ Φ Φ R 1 dφ R dt 4

62

63 60 5 Φ B ds B 5.2 q( E + v B) V = ( v B) l (5.3) v B l l e v B e E E + v B = 0 l 6 V = E l = ( v B) l (5.4) 6 V = E l E l V

64 l V = dφ dt d l v dv = ( v B) d l (5.5) ( A B) C = ( C A) B 7 dv = (d l v) B (5.6) d l v d l v d l v B dv ds B B d S dt a Φ = Ba 2 cos ωt (5.7) cos ωt < 0 V = dφ dt = Ba2 ω sin ωt (5.8) R IV = B2 a 4 ω 2 R I = V R = Ba2 ω R sin 2 ωt (5.10) BIa BIa sin ωt ω ω t sin ωt (5.9) BIa sin ωt aω 2 2 = B2 a 4 ω 2 sin 2 ωt (5.11) R 8 7 A, B, C 8 BIa cos(ωt + α) ω ω t B B

65 v v ev B ev B ev B 5.2 B l I B, I BIl v B I BIlv BIlv = IV (5.12) V = Blv 5.3 F = q( E + v B) 9

66 V = dφ dt V E Φ = ds B ds B S t = E d x (5.13) S 10 S S ds S d x A = S d S (rot A) (5.14) ds B S t = ds (rot E) (5.15) S ds 11 S rot E = B t rot E = 0 V = dφ dt F = q( E + v B) rot E = B V = dφ t dt V = dφ dt V = dφ dt V = dφ dt ω G (5.16) 10 Φ d dt B t B B( x, t) Φ x Φ(t) Φ 11 S 0 0

67 64 5 v e v B V = dφ dt FAQ B t 0 B rot E = 0 V E = grad V rot E = B t E = grad V B = rot A (rot A) t rot A t = rot ( grad V ) E = grad V = 0 (5.17) E = grad V A t (5.18) (rot A) t rot A t = rot ( grad V A t ) = rot A t (5.19) (5.18) B = rot A B A E A t

68 Φ 1 Φ 1 = L 1 I 1 + M 12 I 2 + M 13 I 3 + (5.20) L 1, M 12, M 13, L 1 1 M M 13 [Wb] [A] [Wb/A] [H] 12 Φ 1 = L 1 I 1 + M 12 I 2 + M 13 I 3 + Φ 2 = L 2 I 2 + M 21 I 1 + M 23 I 3 + Φ 3 = L 3 I 3 + M 31 I 1 + M 32 I 2 +. (5.21) i V i = dφ i dt di i = L i dt M di 1 i1 dt M di 2 i2 dt M di 3 i3 dt + (5.22) M ii = L i V i = j M ij di j dt I 1 x B 1 ( x) I 2 ds B 1 I 2 M 21 I 1 = I 2 d S B 1 (5.23) B 1 A I 1 B 1 µ 0 d 3 x j 1 ( x ) ( x x ) 4π x x 3 j 1 ( x ) I 1 x x ( ) 1 x x 3 = x x (5.24) 12

69 M 21 I 1 = = I 2 I 2 ( ds µ0 ( 4π ds ( d 3 x j 1 ( x ) d 3 x µ 0 j 1 ( x ) 4π x x ) ( 1 ))) x x (5.25) x j( x ) A B = B A d 3 x µ 0 j 1 ( x ) 4π x x rot j 1 ( x ) A ds ( rot A( x) ) (5.26) I 2 ds (rot A) I 2 d x A( x) I 2 I 2 M 21 I 1 = d x d x µ 0 j 1 ( x ) I 2 I 1 4π x 2 x } {{ } (5.27) x I 1 j( x ) 0 d 3 x j( x ) I 1 I 1 d x 1 = A( x) M 21 I 1 = µ 0I 1 4π I 2 I 1 d x 2 d x 1 1 x 2 x 1 (5.28) I 1 M 21 = µ 0 1 d x 2 d x 1 4π I 2 I 1 x 2 x 1 M 12 = M 21 I 1 1A I 2 1A (5.29) 1,2 L 1 = M 11 = µ 0 d x 1 d x 1 1 4π I 1 x 1 x 1 (5.30) I 1 x 1 = x 1 j 1 ( x) L 1 (I 1 ) 2 = M 11 (I 1 ) 2 = µ 0 4π d 3 x 1 d 3 x 1 j 1 ( x 1 ) j 1 ( x 1) x 1 x 1 (5.31) 14 Φ 1 N 1, N 2 dφ 1 N 1 dt, N dφ 1 2 dt 13 E = V 14

70 FAQ IV V I J = I 2 R R a b c I I πa 2 I π(c 2 b 2 ) r r H(r) 2πrH(r) r < a a < r < b I b < r < c I I c2 r 2 c 2 b 2 c < r 0 I πa 2 πr2 = Ir2 a 2 I π(c 2 b 2 ) π(r2 b 2 ) = a < r < b H(r) = I 2πr B(r) = µ 0I 2πr l Φ = l 0 dx b a drb(r) = µ 0Il 2π (log b log a) (5.32) I L = µ ( ) 0l b 2π log a l (5.33) ( a)

71 L L di dt I () LI di dt LI di dt dt = 1 2 LI2 + C (5.34) C 0 0 M I 1, I 2 M di 2 dt, M di 1 di 2 MI 1 dt + MI 2 ( MI 1 di 2 dt + MI dt di 1 dt ) di 1 dt = MI 1 I 2 (5.35) dt L i (I i ) M ij I i I j = 1 M ij I i I j (5.36) 2 2 i i j i 1 2 M 12I 1 I 2 M 21 I 2 I 1 16 (5.36) 1 L i (I i ) 2 + M ij I j = 1 I i L i I i + M ij I j = 1 I i Φ i (5.37) j i j i i Φ i = 1 2 I iφ i = 1 2 I i = = i i i i d x A i d S B d S (rot A) i,j B = rot A Stokes i (5.38) d x A Id x jd 3 x 1 d 3 x j 2 = 1 2 d 3 x(rot H) A (5.39) j = rot H = H div ( V W ) = V (rot W ) (rot V ) W (5.40) 16

72 d 3 x(rot H) 2 A = 1 d 3 x H 2 (rot A) 1 d 3 xdiv ( H 2 A) = 1 d 3 x H 2 B (5.41) + ( ) 17 1 d 3 x H 2 B 1 d 3 x D 2 E j A 1 2 j A 1 2 j A 1 2 j A 1 2 j A 1 2 j A z q z = 0 ρ B(ρ, t) t z ρ t = 0 B z B (1) t = 0 (2) ρ Φ(t) B(ρ, t) ρ y (3) B(ρ, t) x (4) B(ρ, t) ρ Φ(t) B(ρ, t) Φ(t) = πρ 2 B(ρ, t) 5-2 Z 17 d 3 xdiv () 0 18 ε 0 2 E 2, µ 0 2 H 2 k 2 x2

73 S 1 S 1 S 2 S 2 S 3 V 5-4 z z I(t) a zx x v I(t) v x x r H H = I(t) 2πr I(t r/c) c H = 2πr r/c

74 rot E = B t (6.1) rot H = j (6.2) 1 div D = ρ, div B = 0 (6.3) rot H = j H ( ) (rot H)z div z z 1 2

75 72 6 div x, y, z z div (rot H) = 0 rot div 0 div (rot H) = 0 div j 0 div j = 0 div div j = ρ t ρ 3 (6.4) -Q +Q H 3 rot E = B t t (div B) div B = 0 0

76 D rot H = j rot H = j div ρ t div D = ρ D t rot H = rot H = j j + D t (6.5) D t (displacemenmt current)5 div D = 0 rot E = B t div B = 0 rot H = 0 (6.6) rot rot D t 6 7 div D = ρ div B = 0 rot E = B t rot H = j + D t (6.7) D = ε 0 E + P, B = µ0 H + M 8 4 D j = ρ t F = q( E + v B)

77 74 6 E, B D, H P, M E, B, P, M E, B, P, M ε 0 div E = ρ div P div B = 0 rot E = B t 1 rot B µ = j + rot M + P 0 t + ε E 0 t ρ div P j + rot M + P t 9 (6.8) D D t 2 D t 2 = 0 z = z = 0 z I I t = 0 Q = It D = It 4πr 2 e r (6.9) t It r 11 D t = I 4πr 2 e r (6.10) ( B( x) = µ j( x ) + ) 0 d 3 x td( x ) ( x x ) 4π x x 3 (6.11) θ A r B P 9 D electric displacement D 10 11

78 P P A B t D( x ) ( x x ) A B 12 x z θ r z H(r, θ) H(r, θ) 2πr sin θ = d S D t I z = 0 I (6.12) z = (1) (2) (rot H = j + t D) (3) (rot E = t B) rot E 0 rot E = B t rot E rot H = D t j

79 76 6 rot E B t E B div E = 0, div B = 0, 1 µ 0 rot B = ε 0 t E, rot E = t B (6.13) B = µ 0H D = ε0e D H E B rot B = ε 0 µ 0 E t rot E = B t rot ( rot rot E ) = ( rot B t ) } {{ } grad ( ) div E } {{ } =0 =ε 0 µ 0 t E E = ε 0 µ 0 2 t 2 E E = ε 0 µ 0 2 t 2 E rot (rot A) = grad (div A) A (ε 2 ) 0 µ 0 t 2 E = 0 (6.14) ( 1 2 ) v 2 t 2 u = 0 v = 1 ε0 µ 0 1 ε0 µ 0 grad rot ( ) div B } {{ } =0 ( rot B ) ( = ε 0 µ 0 rot E t ) } {{ } E = ε 0 µ 0 2 t 2 B B = ε 0 µ 0 2 t 2 B = t B (6.15) π 10 7 = m/s (6.16) 14 z c z ct f(z ct) ( 1 2 ) ( 1 c 2 t 2 2 ) f(z ct) = c 2 t 2 2 z 2 f(z ct) = 1 c 2 c2 f (z ct) f (z ct) = 0 (6.17) m/s (29) (97) (92) (458)

80 E(z ct), B(z ct) div E = 0 div E(z ct) = z E z(z ct) = E z(z ct) = 0 (6.18) z z x, y x E y = 0rot E = B t x y E z z E y = t B x y z 0 = t B x z E x x E z = t B y E x(z ct) = t B y x E y y E x = t B z 0 = t B z (6.19) B x, B z 0 B y (z ct) E x(z ct) = cb y (z ct) z x y 1 c E = (E x (z ct), 0, 0), B = (0, 1 c E x(z ct), 0) (6.20) rot B = ε 0 µ 0 t E x E z y B

81 z = z = I r I 4πr 2 2πrH = I 2 H = 0 H = A A x E z y B

82 79 E-B, 6 E-H, 6, 11, 33, 43, 38, 32, 2, 43, 57, 6, 2, 43, 75, 57, 73, 44, 43, 15, 57, 73, 69, 35, 34, 73, 57, 59, 32

96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A

96 7 1m =2 10 7 N 1A 7.1 7.2 a C (1) I (2) A C I A A a A a A A a C C C 7.2: C A C A = = µ 0 2π (1) A C 7.2 AC C A 3 3 µ0 I 2 = 2πa. (2) A C C 7.2 A A 7 Lorentz 7.1 Ampère I 1 I 2 I 2 I 1 L I 1 I 2 21 12 L r 21 = 12 = µ 0 2π I 1 I 2 r L. (7.1) 7.1 µ 0 =4π 10 7 N A 2 (7.2) magnetic permiability I 1 I 2 I 1 I 2 12 21 12 21 7.1: 1m 95 96 7 1m =2 10 7 N

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