8 電流與磁場、安培定律Magnetic Fields due to Currents

Conventional rocket EM Rail Gun

8-1 Calculating the Magnetic Field due to a current

The law of Biot and Savart

dE dqr

dE dqr

r

dB idsr

dB ids rr

=

=

=

=

14

14

4

4

02

03

02

03

pe

pemp

q

mp

v v

v v

sin

typermeabili m/AT104 70 = pm

=

==

=

0 20

0

20

sin2

2

sin4

rdsi

dBdBB

ridsdB

qp

m

qp

m

Magnetic Field Due to a Current in a Long Straight Wire

Ri

Rss

Ri

RsRdsiB

RsRRsr

pm

pmp

m

q

2]

)([

2=

)(2

=sin ,

002/122

0

0 2/3220

22

22

=

=

=

Integration

0 02 2

020

0 0 00

sin 90 =4 4

4

=4 4 2

ids idsdBR R

iRdB dBR

i i idR R R

m mp p

m p

m m mp p

=

= =

=

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

Ri

RiB

BBrsd

8=

4)2/(

00

003

21

mppm

=

===

例 1 What B does the current produce?

diLiLBiF

BLiFdiB

baabba

abba

aa

pm

pm

290sin

2

0

0

==

=

=

8-2 Two Parallel Currents

0 0 02 2

( ) ( ) ( )

=2 ( ) 2 ( ) ( )

a bB x B x B xi i idd x d x d xm m m

p p p

=

=

例 2 The Field Between Two wires

8-3 Ampere’s Law

encisdB 0m = Comparing Gauss’ law and Ampere’s law

Ampere’s law

irB

rBdsB

dsBsdB

0)2(

)2(

cos

mp

p

q

=

==

=

The Magnetic Field Outside a Long Straight Wire with Current

2

2

0

2

2

)2(

)2(

RrirB

Rrii

rBsdB

enc

ppmp

pp

p

=

=

=

The Magnetic Field Inside a Long Straight Wire with Current

2

4 4

4 40

(2 )

( ) 2

( )(2 )2

r

enc ai JdA cr rdr

c r a

c r aB r

p

p

m pp

= =

=

=

例 3 A hollow conducting cylinder

8-4 Solenoids and Toroids

Magnetic Field of a Solenoid ( 螺線管 )

Magnetic Field of a Toroid ( 螺線環 )

inBinhBh

sdBsdB

sdBsdBsdBa

d

d

c

c

b

b

a

00

mm ==

=

Magnetic Field of a Solenoid

riNBiNrBp

mmp2

)2( 00 ==

Magnetic Field of a Toroid

磁圍阻核融合反應器

Tokamak Fusion Test Reactor JET

8-5 A Current Carrying Coil as a Magnetic Dipole

A current loop and a bar magnet

coil) ofcenter (at 2

)(2

2)(4

4

cos

0

2/322

20

2/3220

20

Ri

zRiR

RzRiRrR

rids

dBB

m

m

pp

mp

m

=

=

=

=

Magnetic Field of a Coil

Coulomb’s Law• Using Gauss’s

law to take advantage of special symmetry situations

• Gaussian surfaces

• 高斯面上各點電場與面內總電荷相關

Gauss’ Law

• Flux enclosed charge

encq

AdE

=

=

0

0

e

e

Gauss’Law and Coulomb’Law• From G.L. to C.L.

20

20

0

00

41

)4(

rqE

qrE

qdAE

qEdAAdE

pe

pe

e

ee

=

=

=

==

敬請期待電磁 IV