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

==退出返回

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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