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# 9 電流與磁場、安培定律

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## 9 電流與磁場、安培定律

Magnetic Fields due to Currents

Conventional rocket EM Rail Gun

The space shuttle

(2)

9-1 Calculating the Magnetic Field due to a current

The law of Biot and Savart (必歐-沙伐定律)

dE dq

r

dE dq

r r

dB ids

r

dB id s r

r

=

=

=

=

1 4

1 4

4 4

0 2

0 3

0

2

0

3

pe

pe m

p

q m

p

v v

v v sin

(3)

ty permeabili

m/A T

10 4

0 = 7

m p

## 

=

=

=

=

0 2

0

0 2 0

sin 2

2

sin 4

r

ds i

dB dB

B

r dB ids

q p

m

q p

m

Magnetic Field Due to a Current in a Long Straight Wire

(4)

R i R

s

s R

i

R s

Rds B i

R s

R R s

r

p m p

m p m

q

] 2 )

[ (

= 2

) (

2

= sin

,

0 2 0

/ 1 2

2 0

0 2 2 3/ 2

0

2 2

2 2

=

=

=

### 

Integration

(5)

A hint of Ampere’s Law

0

0

0

0

0

2 2 B i

R

B R i

B dl i

Bdl i

B d l i

m

pp m

m m

m

=

=

=

=

=

(6)

0 0

2 2

0 0 2

0 0 0

0

sin 90

4 = 4

4

= 4 4 2

ids ids

dB R R

B dB iRd

R

i i i

R d R R

m m

p p

m

m p m  m

p p

=

= =

=

### 

Magnetic Field Due to a Current in a Circular Arc of Wire

(7)

R i R

B i

B B

r s

d

= 8 4

) 2 / (

0 0

0 0

3

2 1

m p

p

= m

=

=

=

v v

(8)

d i LB Li

i F

B L

i F

d B i

b a a

b ba

a b

ba

a a

p m p

m

90 2 sin

2

0 0

=

=

=

=

v v

v

9-2 Two Parallel Currents

(9)

0 0 0

2 2

( ) ( ) ( )

= 2 ( ) 2 ( ) ( )

a b

B x B x B x

i i id

d x d x d x

m m m

p p p

=

=

(10)

9-3 Ampere’s Law

ienc

s d

B m0

### 

v v =

Comparing Gauss’ law and Ampere’s law

Ampere’s law

eenc0

A q d

E =

v v

B B dA

 =

ε =

Ε ds

(11)

i r

B

r B

ds B

ds B

s d B

) 0

2 (

) 2

(

cos

m p

p q

=

=

=

=

### 

v v

The Magnetic Field Outside a Long Straight Wire with Current

(12)

2 2 0

2 2

) 2

(

) 2

(

R i r

r B

R i r

i

r B

s d B

en c

p m p

p

p p

p

=

=

=

v v

The Magnetic Field Inside a Long Straight Wire with Current

(13)

2

4 4

4 4

0

(2 )

( )

2

( )

(2 )

2

r

enc a

i JdA cr rdr

c r a

c r a

B r

p p

p m p

= =

=

= 

###  

(14)

9-4 Solenoids and Toroids

Magnetic Field of a Solenoid (螺線管)

Magnetic Field of a Toroid (螺線環)

(15)

in B

inh Bh

s d B

s d B

s d B

s d B

s d B

a d d

c

c b b

a

0 0

m

m =

=

=

## 

v v v v

v v v v

v v

Magnetic Field of a Solenoid

(16)

r B iN

iN r

B p

m m

p ) 2 2

( = 0 = 0

Magnetic Field of a Toroid

(17)

Tokamak Fusion Test Reactor 1982 - 1997

JET ITER

Joint European Torus

International Thermonuclear Experimental Reactor

(18)

9-5 A Current Carrying Coil as a Magnetic Dipole

A current loop and a bar magnet

(19)

coil) of

center 2 (at

) (

2

) 2 (

4 4

cos

0

2 / 3 2 2

2 0

2 / 3 2 2

0 2 0

R i

z R

iR

z R R

iR r R r

ids dB B

m

m p p

p m m

=

=

=

=

Magnetic Field of a Coil

(20)

conductor pn junction diode pn junction

A diamagnetic material placed in an external magnetic field B ext develops a magnetic dipole moment directed opposite B ext.. If the field is nonuniform, the diamagnetic material

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