chapter 10 magnetic field of a steady current in vacuum

Chapter 10Chapter 10

Magnetic Field of a Steady Current in Vacuum

§ 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

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

N pole S pole

Magnetic monopole ？

N pole S pole

Never be seen!

(2) The magnetic field surrounding the earth

(3)The interaction between current and magnet

N SA BI

N S

S

N

N SI

F

I

attractiattractionon

repellerepellentnt

The motion of electron in M-field

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.

3. Magnetic field3. Magnetic field

M-M-fielfieldd

Moving charge

Moving charge

M- M- fielfieldd

magnet

magnet

current

current

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:

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

Superposition principle of M-fieldSuperposition principle of M-field

i

iBB

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

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

§§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

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

[[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

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

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

[[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

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

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

[[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. .

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 ：：

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..

[[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

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

[[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 ,,

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

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

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

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

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

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

ld

//ld

L

ldB

LldldB )( //

L

Bdl

I0

ldAny Any LL surrounding surrounding II

ld

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

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

[[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 ：：

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

[[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

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

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.

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:

[[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

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

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

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

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.

::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

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

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

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

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

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

M-field of the earthM-field of the earth

Van Allen radiation beltsVan Allen radiation belts beautiful aurorabeautiful aurora

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

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

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

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

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

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

[[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

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

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

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.

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

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

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

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

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

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 ：：

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

=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

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

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