7
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: 1-2 l , L0 C0 , , d m/ dt > 0. pmi (. 8.1) Ii (.8.2) 8.0 " . " B . pmi Ii , (.8.1) Ii Ε i12 (8.6) 8.0 " . " 1-2 " ".
v 1-2 l B (8.6) 8.0 " . " 1-2 , Ε i12 .8.1 : Ε i12 = B v l. (1.1) IL, IC (.8.1) L0 C0 " ". Ε L (8.13) 8.0 " . " L0 (6.39) 6.0 " " UC C0 t : t Ε L = - L0 (dIL / dt); UC = q/ C0 ↔ dUC/ dt = (1/ C0) dq/ dt ↔ dUC/ dt = (1/ C0) IC ↔ UC = (1/ C0)∫ ICdt. (1.2)
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0 Ε i12 (1.1) 1-2 l , , .. Ε i12 I II , (1.2) Ε L UC C0 t II I II (.8.1) " " : t
(I): - Ε i12 = UC ↔ B v l = - (1/ C0)∫ ICdt ↔ d v / dt = - IC/ BlC0 ↔ IC = - BlC0 (d2z/ dt2). (1.3) 0 (II): - Ε i12 + Ε L = 0 ↔ B v l =- L0 dIL / dt ↔ IL = - (Bl/ L0) ∫ v dt ↔ IL = - Blz/ L0 . (1.4)
0
III Ii, IC (1.3) IL (1.4) , C0 L0 , III I III :
(III): Ii + IC + IL = 0 ↔ Ii = BlC 0 (d 2 z / dt 2) + Blz / L 0 (1.5) F , 1-2 l , (7.75) 7.1 " . " d F , 1-2 Ii dl , d l (7.1) 7.1 " . " 1-2 l , : 2 2 2 2 F = ∫ d F = Ii∫ [ d l, B] = Ii∫ [ ( j i/ ji), B] dl = Ii [ ( ji / ji), B] ∫ dl = Ii [ ( ji / ji, B] l, (1.6) 1 1 1 1
ji - (. 8.1) Ii , Ii 1-2 l ; ji / ji - , ji (.8.1) 8.0 " . " Ii 1-2 l Ii .
F OZ , ji / ji B , (1.6) [ ( ji / ji), B] , OZ , B B , II OZ 1-2
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m : OZ: m(d2 z/ dt2) = F Z + mg = - IiBl + mg. (1.7) Ii (1.5) (1.7) 2 - : (d2 z/ dt2) + [ B2 l2/ L0 (m + B2 l2 C0)] z = mg/(m + B2 l2 C0). (1.8) z1(t) (1.8) 2 - 2 - (2.7) 2.0 " " : z1(t) = Acos(ω0 t+ φ0), (1.9) ω0 = [ B2 l2/ L0 (m + B2 l2 C0)]1/2, 1/ - 1-2 ; A; φ0 - .
z2(t) (1.8) 2 - 2 - (2.61) 2.0 " " Ω , β 1-2 : z2(t) = A cos(Ω t+ φ0), (1.10) A = mg/[(m + B2 l2 C0) (cosΩ t)][(ω02 - Ω2)2 + 4β2 Ω2]1/2 = mg/(m + B2 l2 C0) ω02 =
= mgL0(m + B2 l2 C0)/(m + B2 l2 C0) B2 l2 = mgL0/ B2 l2; tgφ0 = - 2β Ω/(ω02 - Ω2) ↔ φ0 = 0 - A φ0 z2(t) (1.10) Ω = 0 β = 0 (1.8) 2 - . z2(t) (1.10) : z2(t) = A ↔ z2(t) = mgL0/ B2 l2, (1.11) .. z2(t) (1.11) (1.8) 2 - (.8.1) 1-2 OZ m g .
z(t) (1.9) z1(t) (1.11) z2(t) (1.8) 2 - :
z(t) = z1(t) + z2(t) = Acos(ωt+ φ0) + A = Acos(ωt+ φ0) + mgL0/ B2 l2. (1.12) dz/ dt 1-2 OZ t (1.12) : dz/ dt = - Aωsin(ωt+ φ0). (1.13) t0 , .. t0 = 0, z0 dz/ dt| t0 = 0
1-2 .
(2.5) 2.0 " "
A φ0 t0 = 0, (1.12), (1.13) :
z0 = Acosφ 0 + mgL0/B2l2 0 = Acos φ 0 + mgL0/B2l2 ↔ φ 0 = π ↔ A = mgL0/B2l2. (1.14)
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v 0Z = (dz/dt) t0 = 0 = - Aωsinφ 0 ↔ 0 = - Aωsin φ 0 (1.13) A φ 0 z(t) dz/dt 1-2 OZ : z(t) = mgL0/B2l2[cos(ωt + π) + 1] ↔ z(t) = mgL0/B2l2(1 - cosωt).
dz/dt = -(ωmgL0/B2l2) sin(ωt+ π) ↔ dz/dt = (ωmgL0/B2l2) sinωt. (1.15)
, (. 08.1.1) 1-2 (1.9) ω0 = [ B2 l2/ L0 (m + B2 l2 C0)]1/2, 1/ (1.12)
A = mgL0/ B2 l2, OZ c (1.14) A = mgL0/ B2 l2 , A .
1
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(8.82) 8.0 " . " , , H (.8.2) l t ND D S , l , H : ∫ H d l = ∂ ND/∂ t = (∂/∂ t)∫ D d S. (2.1) l S
S , (.8.2) a . a A , R O , t0 = 0 q .
, q (.5.8) 5.1 " . . " . D (5.87) 5.2 " . . " , D r - R, O , t q , . D r
D q (.8.2) R (5.89) 5.2 " . . " : ∫ D d S = ∫ D r dS = q ↔ D r4 πR2 = q ↔ D r= q/4 πR2, (2.2) S0 S0
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S0 = 4 πR2 - R , q .
dND D dS (.8.2) (2.2) D r D q : dND = D d S = D r dS = (q/4 πR2)2 πR(sinα′) Rdα′ = (q/2)(sinα′) dα′, (2.3)
dr = R dα′ - dS ; r = R(sinα′) - dS ; α′ - OY , 0 α.
dND D dS (.8.2) ND D S , ND : α α
ND = ∫ D d S = ∫D rdS = (q/2)∫(sinα ′)dα ′ = - (q/2)cosα ′| = (q/2)(1 - cosα), (2.4)
S S 0 0
α - OY ; S - .
t ND (2.4) D S : ∂ ND/∂ t = (∂/∂ t)∫ D d S = (∂/∂ t)[(q/2)(1 - cosα)] = (q/2)(sinα)(dα/ dt). (2.5) S
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dt , q OY dl , R0 - , O , t0 = 0 q . dα α R0 R1 - R (2.6) .
(2.6) t α (2.5) t ND D (.8.3) S , v v OY q α R - , q A , H : ∂ ND/∂ t = (∂/∂ t)∫ D d S = (q/2)(sinα) v sin α/ R = (q v /2 R) sin2α, (2.7) S
R - - R, q A , H .
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(2.1) H (.8.3)
l , a , H l
H H l , : ∫ H d l = 2 πa= 2 πRsinα. (2.8) l
(2.8) H (. 08.1.3) l (2.7) t ND D S , l , H H l , v v OY q α - R, q A l :
∫ H d l = ∂ND/∂t = (∂/∂t)∫ D d S ↔ 2πRsinα = (q v /2R)sin2 α ↔ = q v Rsin α /4πR3. (2.9)
l S
(2.9) H H l H , v (. 08.1.3)
OY q R - , q A l :
= q v Rsin α /4πR3 ↔ = q [ v R] /4πR3 ↔ B = μ 0 q [ v R] /4πR3, (2.10)
B = μ0 - (7.95) 7.2 " " H B .
(2.10) (7.7) 7.1 " . " Bq M , q , v , (v << ).
2
ρ ε . q+ . : ) j j ; ) I , . : ρ; ε; q / j = f(j ) =? I=?
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" . . " , Er, Dr r - E , D , , r . N D D (.8.4)
Sr , (5.118) 5.2 " . . " : N D = ∫ D d S = ∫ Dn dS = ∫ Dr dS = Dr4π r2 = q ↔ Dr = q/4π r2, (3.1) (Sr) (Sr) (Sr)
q - , Sr = 4πr2 .
(8.77) 8.0 " . " j t D : j = ∂ D /∂ t. (3.2) (.8.4) , ρ ε , , D , , - Dr r - .
(3.2) jr r - j : jr = ∂ Dr/∂ t. (3.3) (3.1) Dr r - D (3.3) jr r - j : jr = (1/4π r2) dq/ dt, (3.4) jr - r - j , dq < 0, .. (.6.2) 6.0 " " (.8.4) "" ; jr - r - j , dq > 0, .. (.6.3) 6.0 " " (.8.4) "" .
I (.6.2), (.6.3) 6.0 " " (. 8.4) Sr (6.8) 6.0 " " : I = - dq/ dt, (3.5) (6.19) 6.0 " " j E : j = σ E. (3.6) (.8.4) , ρ ε , , E , , - Er r - . (3.6) j E , j , , - jr r - , (3.4) :
jr = I/ Sr = I/4π r2 = - (1/4π r2) dq/ dt, (3.7) Sr = 4π r2 - , ; jr - r - j , dq < 0, .. (.6.2) 6.0 " " (.8.4) "" ; jr - r - j , dq > 0, .. (.6.3) 6.0 " " (.8.4) "" .
(3.4), (3.7) jr, jr r -
J, j , . ,
j, j , , .. j, j , : j = - j. (3.8)
. 8.4 , q , .. (.6.2) 6.0 " " "" , dq < 0. (3.7) jr r - j , (. 8.4) j r - . (3.8) j (. 8.4) r - .
jr r - j (3.6) : jr = Er/ρ = Dr/ ε0ερ. (3.9)
(3.1) Dr r - D (3.9) jr r - j : jr = q/4π r2 ε0ερ, (3.10)
(3.8) jr r - j (3.9) : jr = - q/4π r2 ε0ερ, (3.11)
ρ = 1/σ - , σ ;
ε - ;
"-" (3.10) jr r - j (.8.4) q ; q jr r - j .
I (6.8) 6.0 " "
(3.11) (. 08.1.4) j Sr= 4π r2 : I = ∫ j d S = ∫ jr dS = - q/4π r2 ε0ερ ∫ dS = - q4π r2/4π r2 ε0ερ=- q/ ε0ερ, (3.12) (Sr) (Sr) (Sr)
jr = - q/4π r2 ε0ερ - jr r - j (3.12) , jr Sr= 4π r2 ; "-" (3.12) I r -
(.8.4) q+ t ;
q I r - .
3
K(x, y, z, t) () , E OXY x, y : E = a(x i + y j)/(x2 + y2), a - . B′ K′(x′, y′, z′, t′) (), v v K(x, y, z, t) OZ . B ′ K′(x′, y′, z′, t′) ?
: E = a(x i + y j)/(x2 + y2); v = v k / B′ =?
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v << c , K′(x′, y′, z′, t′) OZ K(x, y, z, t) , (3.25) 3.0 " "
β = 1/[1 - ( v 2/ c2)]1/2 , .. β ≈ 1.
(4.1) (1.1) 1.0 " " :
B ′ = - (1/c2)[- i ( v a y/(x2 + y2) + j ( v a x /(x2 + y2)] = - (1/c2) [a/(x2 + y2)]( - i v y + j v x) =
= - (1/ c2) { a /[(x ′)2 + (y ′)2]( - i ′ v y ′ + j ′ v x ′) = - (1/ c2) { a /[(x ′)2 + (y ′)2] [ v r ′ ] = [ a/(r ′)2 c2] [ r ′ v ], (4.2) i = i′, j = j′ - , , .. ; x = K′(x′, y′, z′, t′) , y = y′ - K(x, y, z, t) K′(x′, y′, z′, t′) , K′(x′, y′, z′, t′) K(x, y, z, t) OZ .
(4.2) B ′ K′(x′, y′, z′, t′) v v , K′(x′, y′, z′, t′) OZ K(x, y, z, t) , r ′ - , K′(x′, y′, z′, t′) , B ′ .
(4.2) r′ r ′ - , K′(x′, y′, z′, t′) , B ′ ,
x ′, y′ K′(x′, y′, z′, t′) : r′ =[ (x ′)2 + (y ′)2]1/2, (4.3) .. B ′ K′(x′, y′, z′, t′)
O' X′ Y′ (4.3) r′ .