9 電流與磁場、安培定律
Magnetic Fields due to Currents
Conventional rocket The space shuttle
EM Rail Gun
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
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
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
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
=
=
=
=
=
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
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
例 1 What B does the current produce?
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
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
=
=
例 2 The Field Between Two wires
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 dsi r
B
r B
ds B
ds B
s d B
) 0
2 (
) 2
(
cos
m p
p q
=
=
=
=
v vThe Magnetic Field Outside a Long Straight Wire with Current
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
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
= =
=
=
例 3 A hollow conducting cylinder
9-4 Solenoids and Toroids
Magnetic Field of a Solenoid (螺線管)
Magnetic Field of a Toroid (螺線環)
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
r B iN
iN r
B p
m m
p ) 2 2
( = 0 = 0
Magnetic Field of a Toroid
Magnetic fields in a tokamak - the toroidal field is generated by external coils, poloidal by electric current in the plasma
Tokamak
磁圍阻核融合反應器
Tokamak Fusion Test Reactor 1982 - 1997
JET ITER
• Joint European Torus
• International Thermonuclear Experimental Reactor
"JointEuropeanTorus internal" by Image sourced from the EFDA-JET public relations page. Copyright is held by EFDA-JET.. Via Wikipedia
JET
ITER Technical
ITER Aerial
9-5 A Current Carrying Coil as a Magnetic Dipole
A current loop and a bar magnet
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
