Transcript Arbitrary shaped wire I 均匀磁场中任意曲线导体

Chapter 4
Magnetic forces & magnetic fields
磁力和磁场
Magnetic fields in this chapter:
本章研究的磁场
1)In free space, at absence of dielectrics.
在真空中,没有电介质
2) Constant magnetic fields, not varying with time.
恒定磁场,不随时间变化.
Contents in this chapter:
1. The description of M field.
磁场的描述
* 2. the actions of M field to electric current
磁场对电流的作用
(1). The actions of M field to moving charges.
磁场对运动电荷的作用.
(2). The actions of M field to electric currents.
磁场对电流的作用.
(3). The actions of M field to closed current-carrying
conductors. 磁场对闭合载流导体的作用.

Magnetic moment 磁矩
* 3. electric currents produce magnetic fields.
电流产生磁场
The rules of M field.
磁场的规律



Biot-savart law 毕奥-萨伐尔定律
Gauss’ theorem in M field 磁场中的高斯定理
Ampere’s circulation theorem 安培环路定理
§4.1 The M force & M field
1. M field line
M field patterns surrounding a bar magnet as displayed
with iron filings.
M field line
Similar in many ways to the E field lines.
I
I
M lines of current-carrying
straight wire 通电直导线的
M线
M lines of circle current 通电
环导线的M线
I
I
M lines of current-carrying solenoid
通电螺线管的M线
Character of M lines: M线的特征
1)Every M line is a closed curve surrounding the
current. 每一条磁力线都是环绕电流的闭合曲线
But E line is not closed. 电场线不闭合。
since M line is closed, it has no starting point or ending
point. 既然磁力线是闭合的，它没有起点也没有终点。
But E line starts from positive charge and ends at negative
charge. 电场线起于正电荷，止于负电荷。
2) M lines do not cross.
磁力线不相交。
E lines do not cross either. 电场线也不相交。
What is the reason?
*3) The relationship between the direction of M lines and
that of current can be deduced from right-hand rule.
磁力线方向和电流方向之间的关系可以用右手定则来描
述。

2. Magnetic force FB
M lines can describe the patterns of M field and can interpret the strength
of M field quality. But they can not quantify the M field 磁力线可以描
述磁场的式样，也可以定性的说明磁场的强度,但不能定量说明。
E lines
We determine the E field by measuring the E force exerted on a
test charged particle at rest. 我们用施加在静止实验电荷上的电
场力来决定电场大小。
We use an appropriate moving electrically charged particle to
test the strength of M field by measuring the magnetic force
exerted on the test particle. 我们运动电荷来测量磁场强度。
What conditions the particle to test E field must satisfy? 测试电
场的电荷必须满足什么条件？
The moving electrically charged particle to test M
field:
1)The M field produced by the moving charge must be
small enough when compared with the source M field.
该运动电荷产生的磁场与源磁场相比必须足够小。
2)The size of the moving charge must be small enough
that it can be regarded as a particle when it is placed
at a certain point in the free space.
该运动电荷产生的尺寸必须足够小。它放在真空中某一
点时可以被看成是点电荷。


If we put one moving
 test charged particle with v in a M field B ，the
magnetic force FB :

 
FB  qv  B
The direction of the M force is: v
That is perpendicular to both
v
B
and
B
The magnitude of the M force is:
FB  qvBsin 
Where  is the angle between v and B
Cross product
叉乘
FB  qv  B
Three vectors, perpendicular to each
other
Right-hand rule
 右手定则
Fm

B
v

Fm

q


B

v
Point the four fingers of your right hand along the direction of v , then
curl them until they point along the direction of B . The thumb then
points in the direction of FB .


以右手四指由 v 经小于180o的角弯向 B ， 此时，大拇指的指向就
是正电荷所受电场力的方向。

 
FB  qv  B
Note: if the test charge is negatively charged, the direction of
the M force is opposite.
q 
_

Fm

B

v

Fm


q

FB ：磁场力

Fm ：洛仑兹力
右手定则
左手定则

B

v
E force and M force on charged particles
点电荷的电力和磁力
1) The E force is always parallel or anti-parallel to the direction of the
E field, whereas the M force is perpendicular to the M field.电场力总
是平行或者反平行于电场方向，而磁场力总是与磁场方向垂直。
2) The E force acts on a charged particle independent of the particle’s
velocity, whereas the M force acts on a charged particle only when the
particle is in motion.施加在电荷上的电场力与电荷的运动无关，而
只有电荷在运动时，才可能有施加在电荷上的磁场力。
3) The E force does work in displacing a charged particle, whereas the
M force associated with a steady M field does not work when a
charged particle is displaced. 在电场中移动电荷时电力要做功，但
在恒定磁场中移动电荷, 磁场力不做功。
AFB  0
Why？？
Because the M force is always perpendicular to the M field.
磁场力总是与磁场方向垂直。


FB  ds  FB  dv dt  0
AFB  0
According to work-kinetic energy theorem, the kinetic energy of a
charged particle can not be altered by a constant M field alone. 根据功
能原理，单纯的恒定磁场不能改变点电荷的动能。
The applied M field can alter the direction of the velocity vector, but
can not change the speed of the particle.
What is the motion of a charged particle in a constant
M field?
Motion of a charged particle in a constant M field
Bin into the paper
×××××××××
v× ×+q ×F× × × × × ×v
F
B
B
The magnitude of M force is always qvB
The direction of M force always points to the
center of the circle.
× × × × × × × ×+
×
× × × × × × × ×q
×
× × × × ×F× × × ×
B
× × × ×+
×××××
q
v
The particle can be modeled as being in uniform
circular motion.
Newton’s second law:
F  F
B
 ma
mv
The radius of the circular path: r  qB
Cyclotron radius 回旋半径
mv
qvB 
r
2
The angular speed of the particle is
v qB
 
r m
Not depend on the transitional speed of the particle
The period（cyclotron period 回旋周期） of the motion is
2 r 2 m
T

v
qB
Not depend on the radius of the motion
The frequency（cyclotron frequency 回旋频率） of the
motion is
1
qB
f  
T 2 m
If a charged particle moves in a uniform M field with its velocity at
some arbitrary angle to B , its path is a helix.(螺旋线)
y
B
+
q
x
z
Applications of the motion of a charged particle in
M field带电离子在磁场中运动的应用
A charge moving with v in the presence of an E field E and a M field B
experience both an E force and a magnetic force.
  
F  Fe  FB
  
 qE  qv  B
The application of the motion of a charged particle in M filed:
1) Velocity selector 速度选择器
2) The mass spectrometer 质谱仪
3) Cyclotron 回旋加速器
1) Velocity selector 速度选择器
E
qE  qvB v 
B
Bin into the paper
××
××
×＋
× ＋
＋
＋×＋×＋
＋
＋×＋
qv  B
×××××××××
×××
+ ××××××
+
q ××××××
×××
E source
×××××××××
-× -× ×- ×- ×- ×-×--×-×Slit
q
E
qE
Initial velocity V
Only when the E force and M force are in equilibrium, the particle can
pass through the small opening at the end with the initial velocity.
In other words, all the particles passing through the small opening have
the same velocity.
2) The mass spectrometer 质谱仪
Bin
× × × × × × × × ×
× × × × × ×
× × × × × ×
r
× × × × × ×
v
E×
× × × × × × × ×
× × × × × ×
× × × × × ×
q
E
v
B
B0,in
× ×× × × × × × ×
× × × × × ×
× × × × × ×
Velocity selector
× × × × × ×
× × × × × ×
On entering the second M field, the particle moves in a semicircle of
radius r.
rB0 B
m rB0


q
v
E
3) Cyclotron 回旋加速器
Charged particles
accelerated in E field and
do circle motion in M field.
Charges moves inside two
semicircular containers, D1
and D2.
The cyclotron frequency of
the particle equals that of
the alternating potential. 粒
子的回旋频率等于交流电
压的频率。
Gap, the source of particles.
qB
f 
2 m
Move in D1 in circular motion with its frequency
When it arrives the gap, the direction of E field changes, it enters D2 in
circular motion.
After many cycles, the particle arrives
the boundary of the container. At this
time, the cyclotron radius is the radius of
the container R0.
The cyclotron speed of the particle is:
qBR0
v
m
The kinetic energy of the particle is:
2
2
2
0
1 2 qB R
K  mv 
2
2m
When the energy is smaller than 20 MeV, the relativistic effects can
be ignored.
3. Magnetic field

B
FB  qv  B
The magnitude of the M field is:
Fm
Fmax
B
＝
qv sin 
qv
The direction of the M field is:


F v
max
The unit of the M field is: N  s / C  m
or tesla (T)
1T  1N  s / C  m
The M field obeys the superposition principle, like E field.

B 

 Bi
i
§4.2 M force on a current-carrying conductor
载流导体上的磁场力
1) Moving charge in M field
M force on a moving test charged particle:
FB  qv  B
A current is a collection of many charged particles in
motion.
A current-carrying wire also experiences a M force
when placed in an external M field.
One charged particle
One current-carrying wire
+ + + + + + + + +
+
I
V
E field



Q  dQ  dE  E  d E

（segment method 微元法）
2) current-carrying wire in M field
wire segment
Bin into the paper
Suppose the cross area of the wire is A,
×××××××××
and there is n charges per unit volume
×××××××××
The charge in the wire segment is:
×××××××××
×××××××××
×××××××××
×××××××××
I
The wire is
perpendicular to M field
l
ql  nAlq
The M force on the wire segment is:
FBl   nAlq  v  B
And the current in the wire is:
I  nqAv
FBl  Il  B
If the wire is not perpendicular to
M field
F  IBl sin
extend：


l // B

F 0
 
l  B  F  IBl
Straight wire

l

I

B
More generally, an arbitrarily shaped wire of uniform cross
section in an external M field: 横切面积均匀的任意形状的
导线在磁场中的受力
dl : the length segment vector.
I
direction: the same as that of the current
B
dl
dFB  Idl  B
b
FB  I  dl  B
a
Arbitrary shaped wire


B
 
   I B

  
l
The M force of an arbitrary shaped wire in
a perpendicular uniform M field does not
depend on the shape of the wire. It only
depends on the starting and the ending
points of the wire.
在均匀磁场中，垂直于磁场方向的平面
内一段弯曲载流导线受到的总安培力大
小与导线弯曲形状无关，仅与这段导线
起点和终点位置有关。
均匀磁场中任意曲线导体，等于连结曲线始末端的直导线所受
的安培力。
3) Current loop in M field 磁场中的环形电流
I
B
The M forces on sides ① and ③ are zero
because these wires are parallel to the field.
①
②
④ a
×
③
b
A rectangular current loop in
a uniform M field, which is
parallel to the plane of the
loop
The magnitudes of M forces on sides ②
and ④ equal
F2  F4  IaB
But the direction of F2 is out of the paper,
whereas the direction of F4 is into the
paper.
I
The net M forces on the rectangular loop
carrying a current is zero.
①
②
④ a
×
③
b
B
The net M forces in a uniform M field on a loop of any shape
carrying a current is zero. 任意形状的载流闭合线圈在均匀
磁场中的受力为零。
Torque on a current loop in a uniform M field
均匀磁场中载流线圈的转矩
The current loop will rotate if we view from side ③
even though the net M forces on it is zero.
I
①
②
④ a
×
The magnitude of the torque is
 max
③
b
b
 F2  F4  IabB  IAB
2
2
A is the area of the loop.
b
B
F2
②
×④
O
F4
If the M field makes an angle of θto the plane，the net torque
b
b
  F2 sin   F4 sin   IabB sin   IAB sin 
2
2
  IA  B
A vector
Magnitude: the area of the loop.
F2
②
A
B
)θ
×④
Direction: right-hand rule.
When the four fingers of the right hand are curled in the
direction of the current, the thumb points the direction of A
F4
Magnetic dipole moment/magnetic moment
磁矩
Torque:转矩
Definition:
Magnetic moment μ:
moment:力矩
  IA
The torque can be expressed as:
  B
Moment arm:力臂
For any arbitrarily shaped closed current loop in a uniform M field, the
net M force exerted on the loop is zero, the loop will not translate. But
the loop is experienced a torque of     B . The closed loop will
rotate under the torque.任意形状的闭合载流导线在均匀磁场中，所受
的磁场合力为零，线圈不会平动。但会受到一个转矩的作用，闭合
线圈会发生转动。
The effect of the torque is to make the magnetic moment turn to the
direction of M field.力矩的作用效果，总是使磁矩转向外磁场方向。
If the coil consists of N turns of wire, each carrying the same current and
each having the same area, the total magnetic moment of the coil is:如果
一线圈由N匝导线圈组成，并且每匝导线圈有同样的电流和同样的
面积，线圈的磁矩是：
  NIA
Example 4-1
One semicircle wire with radius R, current I
is placed in a uniform M field B. The angle
between the connection of the two ends
and the M field is α=30o. Find the M force
exerted on this arc.
Solution:
The M force on the arc:
 

F  Il B
F  IlB sin
The magnitude of the force :
l  2R
  30o
The direction of the force : into the paper ×

 l
F  IRB

B
§4.3 The Biot-Savart Law
毕奥－萨伐尔定律
We have studied the M forces exerted on 1) moving charges 2) currentcarrying wire 3) current loop by a uniform M field.
In1819, Oersted found an electric current in a wire deflected a nearby
compass needle.
Biot and Savart worked out the relationship between the M field at a
certain point and the current producing the field.

Biot-savart law 毕奥-萨伐尔定律
Biot-Savart law describes the M field dB at point P created by an element
of infinitesimal length dl of the wire.
毕奥－萨伐尔定律描述的是由载流导线电流元在空间中某点产生的
磁场的大小。
Biot-savart law 毕奥-萨伐尔定律
 
 

I d l  er
I dl r
d B  km
 km
2
r
r3

0
7
km 
 10 T  m / A
4
I ：steady current carried by the wire
dl ：an element of infinitesimal length of the wire
er ：unit vector along the vector of r
r ：distance from the element to point P.
I
0 ：permeability of free space真空磁导率

dl

r
P
current element 电流元

I dl
I

I dl

The direction of the current element is the direction
of the current at that point.

r
The magnitude of the current segment is the length
of the wire times the magnitude of current.
大小：导线元的长度乘以电流大小
P
0 Idl sin 
The magnitude of dB d B 
4
r2
The direction of dB
Perpendicular to both dl and
r
I
 
 

I d l  er
I dl r
d B  km
 km
2
r
r3

I dl


r
P
Right-hand rule 右手定则
Point the four fingers of your right hand along the direction of l , then
curl them until they point along the direction of r . The thumb then
points in the direction of dB . 由 Idl经小于180o角转向r 时大拇指所指
的方向就是磁场方向。

dB
Right-hand rule 右手定则

r

Idl
The total M field at some point due to a conductor of finite
size
 


u Idl  r
B  d B＝
4
r

L

L
0
3
M field produced by moving charges运动电荷产生的磁场
The current element produces M field in the space. Actually it
is the moving charges which produce M field.
 
 dB  0 Idl 3 r Idl  qnAv
v
4
r
n：volume charge density
dl
A
电流元
A：area of the current
v ：velocity of the moving charges
0 qnAv  r
dB 
3
4
r



 dB 0 qv  r
B

3
dN 4 r
Thus, the M field produced by moving charges q with the
velocity of v is:


 u qv  r
B
4 r
0
3
The direction of produced M field

r

○
P

v
＋
q0

r

P

v
－
q0
The current element Idl or a moving charge produces a M
field, while the charge element dq produces an E field. 电流
元或运动电荷产生磁场，而电荷单元产生电场。
 

 u0 Idl  r
B   d B＝ 
L
4 L r 3


or B  u qv  r
4 r
0
3
1
dq
E
r
3

4 0 r
Differences:不同点
1) The directions of two fields.
The E field due to a charge element is radial, whereas the M field due to
a current element obeys right-hand rule.
2) The sources of two fields.
An E field can be a result either of a single charge or a charge
distribution, but a M field can only be a result of a current distribution.
Example 4-2
Find the M field in the space around one finite length current
carrying wire.求有限长直载流导线的磁场分布。
I 2


I d l r
0
Reasoning and solution:
2
P

a
1
1
we use Biot-Savart law and segment
method to solve this problem.毕奥－萨
伐尔定律和微元法
 
 0 I d l  r
dB 
3
4
r
I 2
0
 0 I d l sin
dB 
4
r2
r sin  a
a
 
r
I dl
1
direction 
P
l
a
a
r
sin
 ctg 
a d
dl 
sin 2 
0 I 
B
sin  d 

4a 
2
1
0 I
cos 1  cos 2 

4a
The M field due to a finite straight current-carrying wire is:
0 I
cos 1  cos2 
B
4a
the direction of the field can be obtained from right-hand rule.
Discuss:
1) if the wire is infinite, L   , 1  0, 2  
0 I
B
2 a
无限长直载流导线产生的磁场
2) if the wire is half infinite, and the connection line of point
P and the starting point is perpendicular to the wire,
1 

2
, 2  
0 I
B
4 a
半无限长直载流导线产生的磁场
P
3) If point P is at the extending line of the finite length wire,
B  0 延长线上的磁场为零
P
Example 4-3
M field on the axis of a circular current loop 圆电流轴线上的磁场


Idl
I
R

r

d B d B


 B
x
x
d B//
B   d B  0
 0 I sin 
B//  d B// 
2LdlR
2

4r
R
R
sin  
r
R2  x 2

  0 I d l  r
dB 
4 r 3
0 I d l
dB 
4 r 2
d B  d B cos
d B//  d B sin

B

 0 IR i
2

2R x
2
2

3
2
Thus, M field on the axis of a circular current loop is

B

 0 IR i
2

2R x
2
2

3
2

Idl
The magnitude of the M field
B
0 IR
2R  x
2
I
2
2

3
R
2
The direction of the M field, right-hand rule

B
Discuss:
1) If point P is at the center of the circular loop,
0 I
x0 B
2R
In other words, the M field at the center of a completely
circular loop is
  2
0 I
B
2R
For a partial circular loop, the central angle is θ, the M field at
the center of the loop is:
0 I 
  B 
2 R 2

B


I
2) If the distance between P and the plane of the circular loop is
much bigger than the radius of the loop,
x  R
B
0 IR 2
magnetic moment 磁矩：
therefore
B
0 IR
2x
3
2
2x 3
 IA
A: area
0 

3
2 x
A   R2