chapter 10 magnetic field of a steady current in vacuum
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Chapter 10Chapter 10
Magnetic Field of a Steady Current in Vacuum
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§ 10-2 Magnetic Field Gauss’law in Magnetic Field
§ 10-1 Magnetic Phenomena Ampere’s Hypothesis
§10-3 Boit-Savart Law & Its Application
§ 10-4 Ampere’s Law & Its Application
§ 10-6 Magnetic Force on Current-carrying Conductors
§ 10-5 Motion of Charged Particles in Magnetic
§ 10-7 The Hall Effect
§ 10-7 Magnetic Torque on a Current Loop
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1. Magnetic Phenomena
(1) the earliest magnetic phenomena that human knew: the permanent magnet (Fe3O4) has N ,
S poles. Same poles repel each other and different poles attract each other.
§ 10-1 Magnetic Phenomena§ 10-1 Magnetic Phenomena Ampere’s Hypothesis Ampere’s Hypothesis
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N pole S pole
Magnetic monopole ?
N pole S pole
Never be seen!
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(2) The magnetic field surrounding the earth
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(3)The interaction between current and magnet
N SA BI
N S
S
N
N SI
F
I
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attractiattractionon
repellerepellentnt
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The motion of electron in M-field
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2. Ampere’s Hypothesis
Each molecule of the matter can be equated Each molecule of the matter can be equated with a closed currentwith a closed current –called molecular current.–called molecular current.
NS N S
When the molecular currents arrange in same When the molecular currents arrange in same direction, the matter appears magnetism in a direction, the matter appears magnetism in a macroscopic size.macroscopic size.
All Magnetic phenomena result from the motion of the charge.
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3. Magnetic field3. Magnetic field
M-M-fielfieldd
Moving charge
Moving charge
M- M- fielfieldd
magnet
magnet
current
current
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1. Magnetic field
each point in the M-field has a special direction. each point in the M-field has a special direction. when the when the q q moves along this direction( or moves along this direction( or opposite the direction), no force acts on it.opposite the direction), no force acts on it.
The direction of M-field at this point
§ 10-2 Magnetic Field § 10-2 Magnetic Field Gauss’law in Magnetic field Gauss’law in Magnetic field
Take a moving charge( and q) as a test charge.v
The characters of the force on the moving charge by the magnetic field:
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sinqvF
DefinitioDefinitionn sinqv
FB --the
magnitude of M-field
qv
FB max tesla(tesla(TT))oror
M-forceM-force depends on depends on qq,,v v and angle and angle between between and and M-field direction.M-field direction.v
B
along the direction ofalong the direction of vF
max
The The direction of the M-forcedirection of the M-force acting on acting on qq always always perpendicular to and perpendicular to and M-field directionM-field direction..v
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Superposition principle of M-fieldSuperposition principle of M-field
i
iBB
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the magnitude of the magnitude of B
dS
dNB
2. Magnetic field line ( line)2. Magnetic field line ( line)B
tangential direction of line--M-field directiontangential direction of line--M-field direction
..
B
-line is different with line-line is different with line ::B
B
-lines are always closed lines linked with -lines are always closed lines linked with electric current. They have neither origin nor electric current. They have neither origin nor termination.termination.
B
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3. Magnetic flux and Gauss’ Law in magnetics3. Magnetic flux and Gauss’ Law in magnetics
BdSd B SdB
S
Bn
dS
SdBSB
unitunit :: weber(weber(WbWb)T)T·m·m22
For any closed surface For any closed surface S S ,,
S
SdB 0
---- Gauss’ Law in magneticsGauss’ Law in magnetics
----M-field is non-source field
M-fluxM-flux :: The number of -lines through a The number of -lines through a given surface.given surface.
B
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§§10-3 Calculation of the magnetic field set 10-3 Calculation of the magnetic field set up by a currentup by a current
1. Boit-Savart Law 1. Boit-Savart Law
Bd
r
IlIdP
30
4 r
rlIdBd
The magnetic field The magnetic field set up by at set up by at point point PP is is
lId
lId--current element--current element
270 AN104 --permeability of
vacuum
--B-S Law
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Bd
r
IlIdP
For any long current,For any long current,
lBdB
l r
rlId3
0
4
-- superposition principle of M-field-- superposition principle of M-field
2. Application of B-S Law 2. Application of B-S Law
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[[Example 1Example 1]Calculate the M-field of a straight ]Calculate the M-field of a straight wire segment carrying a current wire segment carrying a current II..
I
aP
1
2
l r
solutionsolution
30
4 r
rlIdBd
directiondirection ::
all set up by all have same directionall set up by all have same directionlId
Bd
Choose any Choose any lId
set up atset up at P P ::lId
Bd
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LdBB
L r
dlI2
0 sin
4
ctgal sin
ar
da
dl2sin
B 2
1
sin4
0
da
I
)cos(cos4 21
0
a
I
I
aP
1
2
l r
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discussiondiscussion
for infinite long currentfor infinite long current01
a
IB
2
0
2
)cos(cos4 21
0
a
IB
I
aP
1
2
l r
semi-infinite currentsemi-infinite current
0B
a
IB
4
0
on the prolong line of currenton the prolong line of current
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[[Example 2Example 2] Calculate the M-field on the axis of ] Calculate the M-field on the axis of a circle with radius a circle with radius RR and carrying current and carrying current II..
x0RI
Px
lId
r Bd
Bd
//Bd
SolutionSolution :: choose anychoose any lId
20
4 r
IdldB
//BdBdBd
0 Bd
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LdBB // L
dB sin dlr
IR3
0
4
3
20
2r
IR
23
22
20
)(2 xR
IR
x0RI
Px
r Bd
Bd
//Bd
-------- and and I I satisfy satisfy Right Hand RuleRight Hand Rule B
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discussiondiscussion
at the centerat the center ,, x x =0=0
R
IB
20
0
the magnetic momentthe magnetic moment of the circular current of the circular current
nISpm
23
22
0
)(2 xR
pB m
nRI2
23
22
20
)(2 xR
IRB
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[[Example 3Example 3] Calculate the M-field on the axis of ] Calculate the M-field on the axis of a solenoid with radius a solenoid with radius RR. The number of turns . The number of turns per length of solenoid is per length of solenoid is nn, its carrying current , its carrying current II..
1A 2A
R P1
2
l dl
r
SolutionSolution
The number of The number of turns on length turns on length dl dl ,,
ndldN the current on the current on dldl,, IdNdI
Take Take dldl along axis and along axis and its distance to its distance to PP is is ll. .
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2
222
sin
RlR
1A 2A
R P1
2
l dl
r
2sin
Rddl ctgRl
LdBB
2
1
sin2
0
dnI )cos(cos2 12
0 nI
23
22
20
2 )lR(
dlInRdB
directiondirection ::
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DiscussionDiscussion Solenoid “infinite longSolenoid “infinite long”” ::
1
nIB 0
02 1A 2A
R P1
2
l dl
r
)cos(cos2 12
0 nIB
-- the M-field on the axis of a solenoid with -- the M-field on the axis of a solenoid with infinite length. infinite length.
B
’’s direction: satisfy right-hand rule with s direction: satisfy right-hand rule with II..
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[[Example 4Example 4]A long straight plate of width ]A long straight plate of width LL carrying carrying II uniformly. uniformly. PP and plate current are at and plate current are at same plane.same plane. FindFind B=? B=? of of PP..
LI
dx
dI
SolutionSolution I I dIdI
dxa
IdI
xP d
x
dIdB
2
0
x
dx
a
I
2
0
Straight line currentStraight line current
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dBB
dL
L x
dx
a
I
2
0
L
dL
a
I ln
20
Direction :Direction :
L
P d
dx
dIx
IAll have same direction.All have same direction.Bd
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[[Example 5Example 5]A half ring with radius ]A half ring with radius RR, uniform , uniform charge charge QQ and angular speed and angular speed . Find at . Find at OO..
O
R
SolutionSolution
dldQ
RdR
Q
dQ
It charge,It charge,
?B
dl rx
Take an any Take an any dl dl ,,
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When When dQdQ is rotating, it equates with is rotating, it equates with
T
dQdI
2
dQ2
dI dI set up M-field at set up M-field at OO,,
23
22
20
)(2 xr
dIrdB
dxr
rQ
23
22
2
20
)(2
222 Rxr sinRr
O
R
dl rx
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All All dBdB have same direction have same direction
LdBB
2
0 23
22
2
20
)(2
dxr
rQ
2
0
22
0 sin2
dR
Q
R
Q
80
Direction : Direction :
O
R
dl rx
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3. M-field set up by moving charge3. M-field set up by moving charge
Take Take ,, it set upit set uplId
30
4 r
rlIdBd
30 )(
4 r
rldqnSv
dl
I
v
S
n
The number of moving charges in The number of moving charges in dldl,,
nSdldN
, same direction
lId
v
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30
4 r
rvdNqBd
dN
BdB
30
4 r
rvq
set up by each moving charge ( set up by each moving charge ( qq , ): , ):v
B
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1. Ampere’s Law1. Ampere’s Law
r
IB
2
0
Special exampleSpecial example, infinite , infinite straight line current straight line current II
QuestionQuestion :: L
ldB ?
I
B
§10-4 Ampere’s Law§10-4 Ampere’s Law
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Choose a circle Choose a circle LL is just al is just along ong B-B-line.line.
IB
L
ldB
LdlB L
Bdl
rB 2 I0 L
--does not depend on r
choose is any closed line choose is any closed line L L surrounding surrounding II and and
in the plane perpendicular to in the plane perpendicular to II
L
IldBL 0
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ld
//ld
L
ldB
LldldB )( //
L
Bdl
I0
ldAny Any LL surrounding surrounding II
ld
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a
b1L 2L
11 ldB
I
L L does not surround does not surround II
111 cosdlB drB 11
dI
20
1r
2r
2B1B
d
L
1ld 2ld
22 ldB
drB 22
dI
20222 cosdlB
L
ldB
0 21
21 LLldBldB
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L
IldB 0
--Ampere’s LawConclusionConclusion
NotesNotes:: is the algebraic sum of all currents closed is the algebraic sum of all currents closed
by by LL.. I
I I >0>0 when it satisfy right-hand rule with when it satisfy right-hand rule with LL. .
otherwise, otherwise, I I <0<0..
I I has not contribution to if it is outside has not contribution to if it is outside
LL
lldB
is is non-conservative fieldnon-conservative field..B
Set up by all Set up by all II (inside or outside (inside or outside LL))
Amperian loopAmperian loop
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[[Example1Example1]A long straight ]A long straight wire with wire with RR,uniform ,uniform II. Find . Find BB=?=? inside and outside it.. inside and outside it..
SolutionSolution
R
B
r L
I
L
ldB L
Bdl
Ldr B r B2
2. Application of Ampere’s Law2. Application of Ampere’s Law
Analyze the symmetry of Analyze the symmetry of BB
--Axis symmetry--Axis symmetry
TakeTake L L to be a circle with to be a circle with rr, same direction with , same direction with BB,,
rr>>RR ::
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rr<<RR:the current closed by :the current closed by LL,,
22
rR
II
IrB 02
2
20
R
Ir
R
B
rR020
2 R
IrB
LL
IrB 02
r
IB
2
0
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[[Example 2Example 2] Find the M-field of a infinite ] Find the M-field of a infinite solenoid. The number of turns per length of it is solenoid. The number of turns per length of it is nn, its carrying current , its carrying current II..
R
a
b cd
LldB
daBbcB
nIBB 0
--uniform field
0
exterior : 0B
Choose a rectangular loop Choose a rectangular loop abcdaabcda
SolutionSolution :: analyze the distribution of analyze the distribution of B
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R
O
[[Example 3Example 3] A ] A straight cylinder conductor with straight cylinder conductor with RR.. A A hole with radiushole with radius a a is far is far bb from the central axis of from the central axis of cylinder. The conductor has current cylinder. The conductor has current II ,, Find Find B=?B=? at at point point PP..
SolutionSolution
b
a P
Current densityCurrent density ::
)( 22 aR
Ij
Compensatory method Compensatory method :: Imagine there are and in the hole. Imagine there are and in the hole. j
j
Assume a current Assume a current II in conductor in conductor
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R
Ob
a P
The set by the The set by the conductor conductor with awith a holehole = the set up = the set up by one by one without holewithout hole + + the set up by the set up by the hole’s the hole’s negative negative currentcurrent
B
1B
2B
jbaba
B 201 )(
)(2
jba
2
)(0
1B
Direction :see Fig.Direction :see Fig.
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R
Ob
a P1B
For hole’s For hole’s -j -j ::
jaa
B 202 2
ja
20
Direction: see Fig.Direction: see Fig.
2B
21 BBBP j
aj
ba
2200
)(2 22
0
aR
bI
Direction: Direction:
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[[Example 4Example 4] A ] A conductor flat carries current conductor flat carries current The current density is The current density is jj per unit length along the per unit length along the direction of perpendicular to direction of perpendicular to jj. Find the . Find the distribution of distribution of BB outside the flat. outside the flat.
j
1dl
1Bd
2dl
2Bd Bd
B
P
B
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SolutionSolution
B
B
L
ldB
dabc
ldBldB
Bl2 jl0
20 j
B
At the two side of the At the two side of the flat, M-field has same flat, M-field has same magnitude and magnitude and opposite direction.opposite direction.
a
b c
dl
Take a rectangle path Take a rectangle path abcdaabcda
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1. Lorentz force1. Lorentz force
BvqFm
--Magnetic force acting on --Magnetic force acting on the moving charge.the moving charge.
v //v
vB
mF
§10-5§10-5 Motion of Charged Particles in M-field
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NotesNotes
me FFF
)( BvEq
vFm
②② there are E-field there are E-field ++M-field in the spaceM-field in the space,,
a moving charge a moving charge qq sustains: sustains:
does not do work to does not do work to qq..mF
--Change ’s direction, --Change ’s direction,
don’t change ’s magnitude.don’t change ’s magnitude.
v
v
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Let Let qq goes into M-field goes into M-field with initial velocitywith initial velocityv
2. Moving charge in uniform M-field2. Moving charge in uniform M-field
::Bv
//
q B
--straight line motion with uniform velocity.--straight line motion with uniform velocity.
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::Bv
RO
v
qvBF
qB
mvR
Rvm 2
v
RT
2
qB
m2periodperiod
--Circle motion with --Circle motion with uniform speed in the uniform speed in the plane of plane of B
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Application:Application: mass spectrometermass spectrometer ((质谱仪质谱仪))A charged particle A charged particle
from from SS is speeded is speeded up by up by UU
2
2
1mvUq
Enter M-fieldEnter M-field
UU
SS22
2R2RBB
qB
mvR
(1)(1)
(2)(2)
Combine (1) and (2)Combine (1) and (2)
22
2
RB
U
m
q
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Application:Application: cyclotroncyclotron ((回旋加速器回旋加速器))
qB
mT
2
EE:: speed up speed up qq
BB: : change the change the velocity velocity direction of direction of qq
do not depend on R
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and with any angleand with any anglev
B
cos// vv sinvv
B
v
//vv
---- ---- //// uniform, straight lineuniform, straight line
-------- uniform, circleuniform, circle
Revolving radiusRevolving radiusqB
mvR
helical distancehelical distance
Tvh //qB
mv//2
B
//vv
v
h
Moving pathMoving path------helixhelix
B
B
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Application:Application: magnetic focusingmagnetic focusing ((磁聚焦磁聚焦))
The particles have same The particles have same vv////
B
A
A· ·hh
same same hh
They focus on point They focus on point A again again
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3. 3. Moving charge in non-uniform M-fieldMoving charge in non-uniform M-field
qB
mvR
qB
mvh //2
RR, , hh are different when are different when BB is not constant. is not constant.
AsAs
Magnetic restraintMagnetic restraint ---Magnetic bottle---Magnetic bottle
plasma
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M-field of the earthM-field of the earth
Van Allen radiation beltsVan Allen radiation belts beautiful aurorabeautiful aurora
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is in M-field is in M-field lId
BveFm
nSdldN
§10-6 Magnetic force on a current-carrying conductor
1. Ampere’s Law1. Ampere’s Law
The force acting on The force acting on each electron is each electron is
The numbers of electron in is The numbers of electron in is lId
I
vS
n
B
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The resultant force acting on the The resultant force acting on the dNdN electrons is electrons is
dNFFd m
BlenSvd
BlIdFd
--Ampere’s Law of M-force--Ampere’s Law of M-force
for any shape of current-carrying wire,for any shape of current-carrying wire,
LFdF
LBlId
nSdlBve )(
Ilvddlv
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Take lId
force sinBIdldF
The M-force acting on L is
dlBIdFFL
lsin
0sinBIL
I
L
B
direction :lId
2. The application of Ampere’s Law2. The application of Ampere’s Law
[Example 1] A straight wire with length L carrying I is in a uniform . Find =?B
F
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Set up a coordinate system,Set up a coordinate system,
take any take any lId
BlIdFd
x
y
A B
IFd
yFd
xFd
lId
cosdFdFx
sindF
sinIBdl
IBdy
[[example 2example 2] A curved wire segment with ] A curved wire segment with II is in is in the plane which the plane which . Suppose . Suppose ABAB==L L is is known.Find =?known.Find =?
B
F
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sindFdFy IBdxSimilaSimilar,r,
Vector express:Vector express: jIBLF
Same asSame as the straight the straight wire fromwire from A A to to BB..
l xx dFF
l yy dFF
B
A
y
ydyIB 0
B
A
x
xdxIB IBL
A B
I
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Conclusion Conclusion in a uniformin a uniform , , the M-force acting on any the M-force acting on any
shape wire shape wire == the M-force acting on the the M-force acting on the equivalent straight wireequivalent straight wire . .
B
for a closed wire, for a closed wire, F=0F=0 in uniform in uniform B
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[[Example 3Example 3] ] II11 II22 . . ABAB==L . L . Find =? acting on Find =? acting on
ABAB..
F
II11
II22
dd LL
AA BB
L dl
the force acting on dl is
IBdldF Fd
xx
x
IB
2
10
dFF LBdlI2
Ld
d x
dxII
2210
d
Ldln
II
2210
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d
1I 2I
1B
21Fd
d
IB
2
101
1221 BldIFd
ldI
2
1 1 set up at set up at 2 2 ,,1B
3. The interaction between two parallel currents3. The interaction between two parallel currents
The M-force acting The M-force acting
on , on , ldI
2
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Magnitude Magnitude dlBIdF 1221 dld
II
2
210
directiondirection :: 2211
SimilarSimilarly, ly,
dld
IIF
2
21012
d
1I 2I
1B
21Fd
ldI
2
2B�
12Fd
The force for per unit The force for per unit length wire,length wire,
d
II
dl
dF
2
210
, have opposite , have opposite direction.direction.
21Fd
12Fd
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1. The Hall effect1. The Hall effect
Ib
d B
V
1
2
Experiment result,Experiment result,
HH :: Hall Hall
coefficient.coefficient.
d
IBV HH
§10-7 The Hall effect§10-7 The Hall effect
--there is an electric potential difference on the --there is an electric potential difference on the direction of direction of when a current-carrying plate when a current-carrying plate is put in M-field.is put in M-field.
B
Depends on the material.Depends on the material.
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1
2
VI
B
2. Theoretical explanation2. Theoretical explanation Let the velocity of free electrons is , Let the velocity of free electrons is ,
number density is number density is nnv
v
mF
eF
BveFm
HE
He EeF
0 HEeBve
In equilibrium In equilibrium state,state,
BvEH
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Hall E-P-difference, Hall E-P-difference,
21 UUVH 2
1ldEH
2
1)( ldBv
2
1vBdl vBh
nevbhI
b
IB
neVH
1
neH
1
I
B
1
2
VHE
And And
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For movingFor moving positive positive charges, charges,
1
2
VI
B
v
LF
eFHE
21 UUVH
2
1)( ldBv
2
1vBdl vBh
b
IB
nqVH
1
nqH
1
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NotesNotes ::nn has large magnitude in conductors has large magnitude in conductors
(~(~10102929/m/m33). The Hall effect is not obvious.). The Hall effect is not obvious.
The Hall effect is obvious in The Hall effect is obvious in semiconductor semiconductor n n type semiconductortype semiconductor :: electron conduction.electron conduction. p p type semiconductortype semiconductor :: hole conduction.hole conduction.
to measure to measure HH(or (or VVHH) can judge the moving ) can judge the moving
charges and find charges and find nn..
Positive chargePositive charge
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The normal direction of loop The normal direction of loop ::
B
2l
1l
a
bc
dadF
bcF
II
§10-8 Magnetic torque on a current loop
n
)sin(BIlFad 2
sinBIlFbc 2sinBIl2
Same magnitude, opposite Same magnitude, opposite direction, locate on a line.direction, locate on a line.
1. The torque acting on a loop by M-field1. The torque acting on a loop by M-field
n
Satisfy right hand rule with Satisfy right hand rule with II
0 bcad FF
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cdab FF
Do not locate a line.Do not locate a line.
cdF
abF
B
)(ba
)(cd
n
BIl1
Set up a torqueSet up a torque
sinl
Fsinl
FM cdab 2222
sin21 BlIl
sinISB directiondirection ::
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Vector express:Vector express: BnNISM
cdF
abF
B
)(ba
)(cd
n
Define Define nNISpm
--M-moment of a current loop
BpM m
--can be used for --can be used for any shape plane loopany shape plane loop in in uniform M-fielduniform M-field
For the loop with For the loop with NN turns, turns, sinNISBM
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=0 : MM=0=0
:2
Discussion Discussion
== : : MM =0=0
--stable equilibrium --stable equilibrium position.position.-- unstable equilibrium position-- unstable equilibrium position
When suffers disruption, it turns When suffers disruption, it turns =0=0
BpmmaxMM
The resultant force acting on loopThe resultant force acting on loop=0 =0 in uniform in uniform M-field. But the torque M-field. But the torque 00
--only rotation, not translation--only rotation, not translation
In non-uniform M-field, In non-uniform M-field, MM0, 0, FF 0.0.
--rotation and translation
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2. Potential energy of current loop2. Potential energy of current loop
A current loop hasA current loop has nISpm
II
nB
It suffersIt suffers :: sinBpM m
Bandn M M makes makes decreasingdecreasing
Increase Increase 1 1 to to 22 ,, external force external force does work:does work:
2
1
MdW
2
1
sin
dBpm
21 coscos Bpm
’’s direction:s direction:M
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A loop with M-moment is put in , theA loop with M-moment is put in , the
potential energy of the potential energy of the systemsystem ( ( loop + M-fieldloop + M-field) is) is
mp
B
cosBpW mm Bpm
== The increment of potential energy of the The increment of potential energy of the loop in loop in B
12 mm WW