This document discusses the magnetic force on current-carrying conductors in several situations: 1) It explains Ampere's force law and how magnetic fields exert forces on currents. Magnetic fields inside and outside a current element are the same. 2) It shows examples of calculating the magnetic force on a finite wire, half-circle wire, and rectangular current loop using the right hand rule and Ampere's force law. 3) It discusses how no force is exerted between a straight wire passing through the center of a circular loop, both carrying current, due to the magnetic fields canceling out.

1. 第9章 稳恒磁场
§9.1 磁场 磁感应强度
§9.2 毕奥-萨伐尔定律 磁高斯定理
§9.3 安培环路定理
§9.4 带电粒子在磁场中的运动
§9.5 磁场对载流导线的作用
§9.6 电磁场的相对论性变换

2. Magnetic forces on currents
§9.5 磁场对载流导线的作用
B
r
v
r
lId
r
S
θ
Bvqf
rrr
×=a charge carrier：
在电流元的范围内外磁场 的值相同B
r
磁场对载流导线的作用实际上是载流导线中作定向运动的
大量自由载流子受到洛伦兹力的宏观表现
)( BvqnSdlFd
rrr
×=lId
r
vnqJ
rr
= SJI =
)( BJSdlFd
rrr
×= BlId
rr
×= Jld
rr
//
BlIdFd
rrr
×=
安培力
Ampère force 退出返回

3. BlIdFd
rrr
×= Ampère force 安培力
∫ ×=
L
BlIdF
rrr
Magnetic force on finite wire with
current
3
0
4 r
rlId
Bd
rr
r ×
=
π
µ
2
12
1211220
12
)(
4 r
rˆldIldI
Fd
××
=
rr
r
π
µ
Ampère Law
退出返回

4. )()( iˆjˆBdlI ×=
∫−=
L
dl)kˆ(IB
kˆIBLF θsin−=
r
kˆIBLF −=
r
finite wire with current
BlIdFd
rrr
×=
返回 退出
)(( iˆB)jˆdlIFd ×=
r
)()( kˆBdlI −=
∫ −=
L
kˆdlBIF )(
r
)(sin)( kˆBdlIFd −= θ
r

5. uniform magnetic field
A half-circle wire of radius R carries a current I. The wire is placed
in a uniform magnetic field. The angle between the path from one
end a to another end b of the wire and the magnetic field is α=30
o
.
What is the magnetic force on the wire?
infinitesimal wire with current
B
r
a
b
lId
r
I
lId
r
Rab 2=
α
∫ ×=
ab
BlIdF
rrr
BldI
ab
rr
×⎥
⎦
⎤
⎢
⎣
⎡
= ∫ BabI
r
×=
direction： F
r
⊗αsinBabIF =
r
IBR=
退出返回

6. A long straight wire passes through the centre of circular
flat loop of wire vertically, both carrying current I. what
is the interactive force?
BlIdFd
infinitesimal wire with current
rrr
×=
0=F
r
0=× BlId
rr
lId
r
∫ ×=
L
BlIdF
rrr
An arbitrarily shape flat loop of wire, carrying current
I, is placed in a uniform magnetic field.
∫ ×=
L
BlIdF
rrr
BldI
L
rr
×⎥
⎦
⎤
⎢
⎣
⎡
= ∫ 0=
I
I
Idl
返回 退出

7. Magnetic force on current loops
A rectangular loop of wire, carrying a current and free to rotate
about a fixed axis, is placed in a magnetic field. Magnetic forces
on the wire produce a torque that rotates it.
退出返回

8. a b
θsin31 BIaFF ==
BIbFF == 42
ϕϕ sin
2
sin
2
22 F
a
F
a
M +=
N turns
NISm
mB
NBISM
=
=
=
ϕ
ϕ
sin
sin
BmM
rrr
×=
0=F
r
nˆ
ϕ
ϕ
θ
返回 退出
ϕsinBIS=ϕsinBIba= ϕsinmB=

9. An arbitrary shape flat loop of wire , carrying current
I through uniform magnetic field.
0=F
r
I
Infinitesimal rectangular loops of wire,
carrying current I
BmdMd
rrr
×=nˆIdSmd =
r
∫∫ ×== )( BmdMdM
rrrr
BmM
rrr
×=
)( BnˆIdS
r
×= ∫ BnˆdSI
r
×= ∫ )(
Bm
rr
×=BnˆIS
r
×=
退出返回

10. CAI
退出返回