phy b11 1-2
Post on 11-Aug-2015
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Motional emf§11.2 动生电动势
• Created by a conductor moving in a magnetic field
• “Loop” is imaginary but real emf is induced in conductor
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BlvtB =Φ Blvdt
d B =Φ
Blvdt
d B −=−=ΦεMotional emf :
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ldBvqq
ldELL
k
rrrrr⋅×=⋅ ∫∫ )(1 dlvB
L∫−= Blv−=
A constant magnetic field of magnitude B
BvqFrrr
×=
×××
×××
×××
Br
vrldr
a
bBv
qFEk
rrr
r×==
q > 0
ldBvldEk
rrrrr⋅×=⋅ )(
∫ ∫ ⋅×=⋅=b
a
b
ak ldBvldE
rrrrr)(ε ab
∫ ⋅×=L
ldBvrrr )(ε
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∫ ⋅×=L
ldBvrrr )(ε对于导体回路 L
In constant magnetic field, the motional emf :
dtdldBv m
L
Φε −=⋅×= ∫rrr )(
证明在稳恒磁场中上式是一个普遍规律
In , t∆ ldr
the area swept by )(tc )( ttc ∆+′
ldr
ld ′rv
rtvldSd ∆
rrr×=
0=⋅∫∫Σ
SdBrr
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0)( =⋅+×⋅+⋅−=⋅ ∫∫∫∫∫∫∫′ ScS
SdBtvldBSdBSdBrrrrrrrrr
∆Σ
⎟⎟⎠
⎞⎜⎜⎝
⎛⋅−⋅−= ∫∫ ∫∫
′ S S
SdBSdBrrrr
∫ ×⋅c
tvldB )( ∆rrr
m∆Φ−=
)()()( βαγαγβγβαrrrrrrrrr
×⋅=×⋅=×⋅ )(tc )( ttc ∆+′
ldr
ld ′rv
r
tldBv m
c ∆∆Φ
−=⋅×∫rrr
)(
∫ ×⋅c
Btvld )( rrr
∆ m∆Φ−=
0→t∆ ∫ −=⋅×=dt
dldBv mΦεrrr )(
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?=ε.,,bOA,a,I θω =1. Constant current
( )θπµ
coslaIB
+=
20
I a θωld
r
O
A
ldBvdrrr
⋅×= )(ε l
vBdl=
( )dlcoslaIl
θπµ
ω+
=2
0
⎟⎠⎞
⎜⎝⎛ +
−== ∫ acosbaln
cosab
cosId
b θθθπ
ωµεε
20
0
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2. A copper disk spinning in a uniform magnetic field
∫ ⋅×=a
oldBvrrr )(ε
∫=r
Bdll0ω
Br ω221
= ao →
parallel connection Br ωε 2
21
=
ωO a
ldr
The Faraday Disk Generator
By Ohm’s law the current is RBrI
2
2ω=
Br
ω rI
+ at the rimat the center−
并联
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∫=r
o
vBdl
l
or
In , t∆ the area swept by the conductor rod
o
a
bθ∆θ∆∆ 2
21 rS = r
θ∆∆∆Φ 2
21 BrSBm ==
Loop: obao
dtd mΦ
ε −=direction： ao →
ω221 Br−=
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Br
, the loop of radius R，vr3. Uniform magnetic field
moves with speed
×××
×××
×××
rva
b
c
dθ
（１）The distribution of emfldr
（２） of a and ckEr
Bvrr
×（３）emf between b and d
ldBvdrrr
⋅×= )( )1( ε
θθRdvBsin=
BvEk
rrr×= )2( vBEE kcka == directed to the left
vBRdvBRldBvb
d2sin)( )3(
0==⋅×= ∫∫
π
θθεrrr
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Forces and Energy in Motional emf
Br
∫=L
vBdl
∫
∫⋅×+++=
⋅×=
L
L
ldBv
ldBv
rrr
rrr
)(000
)(ε
vBL=
均匀磁场
××
vrf
r× × × ×× × × ×× × × × ×
× × × × ×I
RBLv
RI ==
ε The magnetic forces on the conductoract to slow down the conductor
∫ ×= BlIdfrrr
RvLBILBf
22
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Force tends to slow the conductor down (drag force)
××
vrf
r× × × ×× × × ×× × × × ×
× × × × ×I
RvLBf
22
=
22 LBmR
=τ
tmR
LBvv dd 22
−=τtd
−=tvm
RvLB
dd22
=−
τt
evv−
= 0∫∫ −=tv
v
tvv
0
dd
0τ
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τt
evv−
= 0 22 LBmR
=τ
The rate at which you do work on the conductor as you pull it from the magnetic field:
RvLBfvP
222
==
The rate at which thermal energy appears in the conductoras you pull it along at constant speed
RvLBR
RBLvRIP
22222 )( ===′
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××
vr
2fr× × ×
1fr
×× × × ×× × × × ×
× × × × ×ur vu
rr+
Fr洛伦兹力做功？
杆以速度 向右运动时vr
BuvqFrrrr
×+= )(
带电粒子所受总的洛仑兹力
BuqBvqrrrr
×+×= 21 ffrr
+=
uBvquf rrrrr⋅×=⋅ )(1 vuBq rrr
⋅×= )( vBuq rrr⋅×−= )(
vf rr⋅−= 2
兹力在这里起到了传递能量的作用。
外力克服洛伦兹力的一个分力 所做的功率通过另一个
分力 做正功全部转化为感应电动势提供的功率，洛伦2fr
1fr
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× × × ×× ×
××××××
× × × ×vr
Fr
× × × ×× ×
××××××
× × × ×vr
Fr
Eddy Currents涡电流
Eddy currents: produced in conductor moving in magnetic field
• Can make a very effective brake
• If you don’t want a brake, eddy currents can be foiled by cutting holes (slots) in conductor
电磁阻尼
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Eddy Currents
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To inhibit the development of eddy currents in the moving metal plate, slots can be cut in the plate
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