Magnetostatics

MagnetostaticsMagnetostatics

Chapter 5Chapter 5

2

The Lorentz Force Law The Lorentz Force Law

EQF

Magnetic Magnetic FieldsFields

3


Magnetic Magnetic FieldsFields

EQF

4


Magnetic Magnetic ForcesForces )Bv(QFmag

)]Bv(E[QFFF magetotal

=> Lorentz Force

5


Exp 1: Exp 1: cyclotron cyclotron motionmotion

pmvQRBR

vmQvB

R

vm)Bv(Q

FF cL

2

2

Bv

Bv

6


Exp 2: Exp 2: A more exotic trajectory occurs if we include a uniform A more exotic trajectory occurs if we include a uniform electric field, at electric field, at right angles to the magnetic one. Suppose, for instance, that right angles to the magnetic one. Suppose, for instance, that BB points in the points in the x-direction, and x-direction, and EE in the z-direction. A particle at rest is in the z-direction. A particle at rest is released from the released from the origin; what path will it follow?origin; what path will it follow?

7


Exp 2:Exp 2:

frequency cyclotron: m

QB

)( )yB

E()y

B

E(

m

QBz

)( zzm

QBy

zmyQBQE

ymzQB

)zzyy(mam)zyByzBzE(Q

)BvE(QFL

2

1

zyByzB

B

zy

zyx

Bv

zzyyxrdt

dv

z)t(zy)t(yxr

00

0

0

0

no force on x-no force on x-directiondirection

8


Exp 2:Exp 2:

321

222

1

2

1

ctB

Etsinctcosc)t(y

B

EtsinBtcosAy

B

Eyy )y

B

E(zy

)(

)( )yB

E(z

)( zy

Substitute to (1)Substitute to (1)

B

Ec, ctsinctcosc)t(z

zB

E)tsinctcosc( zyzy

4412

12

9


Exp 2:Exp 2:The particle start from rest at origin000000 )(z)(y and )(z)(y

B

Ecc,cc

c )(z

cc )(zB

Ec )(y

cc )(y

orf

4231

1

42

2

31

0

000

000

000

000

ctsinctcosc)t(z 412

321 ctB

Etsinctcosc)t(y

)tcos(B

E)t(z

)tsint(B

E)t(y

1

10


Exp 2:Exp 2:

formula circle, R)Rz()tRy(

)tcost(sinR)Rz()tRy(

tcosR)Rz(

tsinR)tRy(

tcosRRz

tsinRtRy

)tcos(B

E)t(z

)tsint(B

E)t(y

222

22222

1

11


CurrentCurrent:the charge per unit time passing a given point:the charge per unit time passing a given point)s(ondsec/)C(coulombs )A(amperes

dt

dQI 11

)tv(dQ

v t

)tv(

dt

dQI

vv

12


CurrentCurrent

d)BI(d)Bv(dq)Bv(F)Bv(qF magiii

imag 1

)Bd(I)Bd(IF.const is I if

mag

13


Exp 3: Exp 3: A rectangular loop of wire, supporting a mass A rectangular loop of wire, supporting a mass mm, hangs , hangs vertically with one end in a uniform magnetic field vertically with one end in a uniform magnetic field BB, which points , which points into the page in the shaded region of figure. For what currentinto the page in the shaded region of figure. For what current I I, in , in the loop, would the the loop, would the magnetic force magnetic force upward exactly upward exactly balancebalance the the gravitational force gravitational force downward?downward?

)Bd(IBvqFmag

mgFg

14


Exp 3:Exp 3:

Ba

mgI mgIBa

mgF

IBa)Bd(IF

g

mag

m

15


Surface current density Surface current density

d

IdK

velocity:v

density eargch surface:, v K

da)BK(da)Bv(dq)Bv(Fmag

16


Volume current density Volume current density

da

IdJ

velocity:v

density eargch volume:, v J

d)BJ(d)Bv(dq)Bv(Fmag

17


volume d)BJ( d)Bv(

surface da)BK(da)Bv(

line d)BI(d)Bv(

ointp Bvq

dq)Bv(F

n

iii

mag

1

SummarizeSummarize

18


Exp 4(a): Exp 4(a): A current A current I I is is uniformly distributed uniformly distributed over a wire of over a wire of circular cross section, with radius circular cross section, with radius aa. Find the volume current density . Find the volume current density J .J .

2a

IJ

19


Exp 4(b): Exp 4(b): suppose the current density in the wire is proportional to suppose the current density in the wire is proportional to the distance from the axis.the distance from the axis. J=ks (k=constant) J=ks (k=constant) . Find the total . Find the total current current II in the wire. in the wire.

3

2

2

3

0

2

2

0 0

ka

dssk

)sdsd)(ks(

JdaI

JdadI

da

dIJ

a

s

20


Equation of continuityEquation of continuity

tJ

d)t

(ddt

dd)J(adJ

VVVS

total charge per unit time leaving a total charge per unit time leaving a volume Vvolume V

S

JdaI e- e- e-e-

e-e- e-e-J

21

The Biot-Savart Law The Biot-Savart Law

The magnetic field of a steady currentThe magnetic field of a steady current

R

RdId

R

RI)r(B

20

20

44

Biot-Savart Biot-Savart lawlaw27

0 104 A/N permeability of free permeability of free spacespaceunitsunits)mA/(N )teslas(T 11

d

R

rPP

22


Exp 5: Exp 5: Find the magnetic field a distance Find the magnetic field a distance ss from a long straight from a long straight wire carrying a steady current wire carrying a steady current II . .

d

R

xyz )sin(sin

s

Idcos

s

I

z dcos)cos

s)(

s

cos(

IB

s

cos

RcosRs

dcos

sdsecsdtans

z cosdz sindRd

R

RdI)r(B

1200

22

20

2

2

2

22

20

44

4

1

4

2

1

2

1

23


Exp 5:Exp 5:

z s

Iz )(

s

I

z )sin(sins

I B

,

22

4

4

22

00

120

12

For an infinite wireFor an infinite wire

24


25


Exp 6: Exp 6: Find the magnetic field a distance Find the magnetic field a distance zz above the center of a above the center of a circular loop of radius circular loop of radius a a, which carries a steady current , which carries a steady current II . .

z )za(

aI

z )R

a(

Ia

z a)R

cos(

I

z cosR

adI)z(B

ˆ R

dI)(B

ˆdRd

R

RdI)r(B

/ 2322

20

30

20

2

0 20

20

20

2

2

24

4

4

4

d

a

R

z

26


volume dR

RJ

surface adR

RK

line dR

RI

)r(B

20

20

20

4

4

4

27

The Divergence and Curl of B The Divergence and Curl of B

I dI

sd s

I

)zdzˆsdsds()ˆ s

I(dB

0

2

0

0

0

0

2

2

2

28


JB

adJad)B(

adJIdB enc

0

0

00

from stoke’s theoremfrom stoke’s theorem

Ampere’s law Ampere’s law in differential formin differential form

Ampere’s law Ampere’s law in integral formin integral formencIdB 0

29


),,(

),,(

ˆ)(ˆ)(ˆ)(

ˆ)(

4)(

2

zyxoffunctionaisJ

zyxoffunctionaisB

zdydxdd

zzzyyyxxxrrR

where

dR

RrJrB o

x

y

z

O r

r R

from Biot-Savart law

d])R

R(J)J(

R

R[

d)R

RJ()r(B

o

o

22

2

4

4

21 p @ )iv.(eq ,)B(A)A(B)BA(

30


])ˆ

()(ˆ

[4

)(22

d

R

RJJ

R

RrB o

)72.1.(42, 0ˆ)(1ˆ)

sin

1(

1ˆ

0),,()ˆˆˆ(

222

eqpRR

RRR

R

zyxJzz

yy

xx

J

since

0)( rB

The divergence of the magnetic field is zero

31


21@).()()()()()( p vieq ,ABBABAABBA

x

y

z

O r

r R

d

R

RJrB o )

ˆ(

4)(

2

22

22222

ˆ)()

ˆ(

)(ˆ

)ˆ

(ˆ

)()ˆ

()ˆ

(

R

RJ

R

RJ

JR

R

R

RJ

R

RJJ

R

R

R

RJ

0 0

dR

RJd

R

RJrB oo

22

ˆ)(

4)

ˆ(

4)(

32

)100.1 .(50, )ˆ(4ˆ

32

eqpRR

R


dR

RJd

R

RJrB oo

22

ˆ)(

4)

ˆ(

4)(

)(

)(4)(4

)ˆ

(4

32

rJ

drrrJdR

RJ

o

oo

x

y

z

O r

r R

since

33

3

2

2/3222

2/3222

2222222

ˆ)(ˆ)(ˆ)()(

ˆ)(

])()()[(

ˆ)(ˆ)(ˆ)()]ˆˆˆ(),,([

])()()[(

ˆ)(ˆ)(ˆ)()]ˆˆˆ(),,([

)()()(

ˆ)(ˆ)(ˆ)(

)()()(

1)]ˆˆˆ(),,([

ˆ)(

R

zzzyyyxxxJ

R

RJ

zzyyxx

zzzyyyxxxz

zy

yx

xzyxJ

zzyyxx

zzzyyyxxxz

zy

yx

xzyxJ

zzyyxx

zzzyyyxxx

zzyyxxz

zy

yx

xzyxJ

R

RJ


21 )(, )()()( piiieqfAAfAf

svv x

xx

x

adJR

xxdJ

R

xxd

R

RJ

JR

xx

R

RJ

R

RJ

JR

xxJ

R

xxJ

R

xx

R

xxJ

R

RJ

332

322

33332

)(]ˆ

)[(

)(]ˆ

)[(]ˆ

)[(

)())(()())((]ˆ

)[(

0 : for steady current: for steady current0 : for surface s → ∞: for surface s → ∞

0ˆ

)(4 2

dR

RJo

34


x

y

z

O r

r R

)(ˆ

)(4

)ˆ

(4

)(22

rJdR

RJd

R

RJrB o

oo

Ampere’s law Ampere’s law –in differential form

35

Application of Ampere’s law Application of Ampere’s law

enco

encoo

o

IdB

IadrJdBadB

rJrB

)()(

)()( Ampere’s law Ampere’s law (in differential (in differential form)form)

Ampere’s law Ampere’s law (in integral (in integral form)form)

law sAmpere law tvarSaBiot ticsMagnetosta

law sGauss law coulomb ticsElectrosta

':

':

36

Comparison of Magnetostatics and Electrostatics Comparison of Magnetostatics and Electrostatics

name no E

law sGauss E

,0

',1

0

law sAmpere JB

name no B

',

,0

0

law Force )BvEQ(F ,

37

Exp 7: Exp 7: Find the magnetic field a distance Find the magnetic field a distance ss from a long straight from a long straight wire, carrying a wire, carrying a steady current steady current II..


ˆ2

2

2)ˆ()ˆ(2

0

s

IB

s

IB

IBsdBssd B

IdB

o

o

o

enco

38

Exp 8: Exp 8: Find the magnetic field of an infinite uniform surface Find the magnetic field of an infinite uniform surface current current , flowing over the , flowing over the xyxy plane. plane.


xKK ˆ

0,ˆ2

0,ˆ2

2

2

)ˆ(ˆ

)ˆ()ˆ()ˆ()ˆ(

0

0

0

z for yK

z for yK

B

KB

KB

dyKdydzxxK

dy yyBdy yyB

adKIdB

o

o

o

o

oo

oenco

39

Exp 9: Exp 9: Find the magnetic field of a very long solenoid, consisting Find the magnetic field of a very long solenoid, consisting of of nn closely closely wound turns per unit length on a cylinder of radius wound turns per unit length on a cylinder of radius RR and and carrying a carrying a steady current steady current II . .


40


41

Exp 9:Exp 9: Application of Ampere’s law Application of Ampere’s law

02 encoIsBdB

42


0)()(

0)()()(

0)]()([

:1

bBaB

B equal bBaB

ILbBaBdB

loop for

also

enco

znIB

nIB

nILIBLdB

loop for

o

o

oenco

ˆ

:2

solenoid the outside

solenoid the inside z nIB o

,0

,ˆ

43

Exp 10: Exp 10: A toroidal coil consists of a circular ring, or ‘donut’, A toroidal coil consists of a circular ring, or ‘donut’, around which a around which a long wire is wrapped. The winding is uniform and tight long wire is wrapped. The winding is uniform and tight enough so that enough so that each turn can be considered a closed loop. The cross-each turn can be considered a closed loop. The cross-sectional shape of sectional shape of the coil is immaterial. I made it the coil is immaterial. I made it rectangularrectangular in in figure a figure a for the sake of for the sake of simplicity, but it could just as well be circular or even simplicity, but it could just as well be circular or even some some weird weird asymmetrical formasymmetrical form, as in , as in figure bfigure b, just as long as the , just as long as the shape remains the shape remains the same all the way around the ring. same all the way around the ring. In that case it follows In that case it follows that the that the magnetic field of the toroid is circumferential at all magnetic field of the toroid is circumferential at all points, both points, both insideinside and and outsideoutside the coil. the coil.


(a)(a) (b)(b)

R

44


From Biot-Savart lawFrom Biot-Savart law

zIyIxIzIsII

zzzysxsx

zzysxszzyxx

rrR

where

dR

RIBd

zsszs ˆˆsinˆcosˆˆ

ˆ)(ˆsinˆ)cos(

)ˆˆsinˆcos()ˆˆ0ˆ(

4 30

45

zxIyzzIsxIxsIzzI

zsxs

II

yzzsx

II

xzzs

II

zzssx

III

zyx

RI

sszzs

ss

zs

zs

zss

ˆ]sin[ˆ)](cos)cos([ˆ)])(([sin

ˆ)cos()sin(

cossin

ˆ)()cos(

cos

ˆ)()sin(

sin

)()sin()cos(

sincos

ˆˆˆ


R

46

y)]zz(cosI)cossx(I[dR

}z]sinxI[

y)]zz(cosI)cossx(I[

x)]sI)zz(I({[sindR

dR

RIBd

sz

s

sz

zs

30

30

30

1

4

1

4

4


B

R

sin)sin(

47


coil the outside points for,

coil the inside points for, ˆs

NIB

ˆs

NIB

s

NIB

NIsB

NIˆsdˆB

IdB

law s'Ampere from

o

oo

o

o

enco

02

22

2

2

0

Ampere’s Ampere’s looploop

48


49

Magnetic Vector Potential Magnetic Vector Potential

potential vector:A ,AB B

potential scalar:V, VE E

0

0

JA

0A since

JA)A()A(B

)A(B

0

0

2

2

0

50


dR

)r(J

4(r)A

JA

0solution

0

2

dR

V

equation s'Poisson ,V

solution

0

0

2

4

1

D, dR

I

4

D, adR

K

4

D, dR

)r(J

4

(r)A

0

0

0

1

2

3

51

Magnetic dipole of the vector potential Magnetic dipole of the vector potential

A

m

I

x

y

z

)z,,x(P 0

a

d

r

r r

)y(ˆ cosr

adIˆr

adI

R

dIA

2

0

02

0

00

444

cosr

ax

r

a

cosr

a

r

a

rrr/r,ra, cosr

ar

r

a

)cosr

ar

r

a(

r

r

cosararr

/

22

2

2

2

22

2

2122

2

222

21

21

12

1

21

2

cosr

xcos cosxacosra

)ysinaxcosa()zzxx(ar

52


A

m

I

x

y

z

)z,,x(P 0

a

d

r

r r

)rm(r

ˆ )(sinr

m ˆ )(sin

r

)S(I

ˆ )r

x(

r

)a(I ˆ )(

r

xIa

ˆ)sin(r

xIa

ˆ dcosr

xIa

ˆ d)cosr

ax

r

a(cos

r

aIA

20

20

20

2

20

3

20

2

03

20

22

03

20

22

22

0

0

4

44

44

24

1

24

4

21

4

dipole magnetic, SIzISm

00

53


A

m

I

x

y

z

)z,,x(P 0

a

d

r

r r

ˆ sinr

m

r cos

r

m

ˆˆ)]sinr

m

rr(

r)sinr

m[(

sinr

AsinrrAAr

ˆsinrˆrr

sinr

AB

ˆsinr

m)rm(

r A

r

30

30

20

22

02

2

20

20

4

2

4

04

4

1

1

44

0 0

54


Exp 11: Exp 11: A spherical shell, of radius A spherical shell, of radius DD, carrying a uniform surface , carrying a uniform surface charge charge σσ, is set spinning at angular velocity , is set spinning at angular velocity ωω. Find the vector . Find the vector potential it produces at point potential it produces at point rr..

D

55


Exp 11:Exp 11:

]z)sinsin(sin

y)cossincossin(cos

x)sinsin(cos[D

cosDsinsinDcossinD

cossin

zyx

)r(

vK

0

)ddsinD(R

)r(Kad

R

)r(K)r(A

200

44

R

DD

0

2

0

2

0dcos dsin

56


Exp 11:Exp 11:R

DD

1

1

2222

2230

1

1 22

30

0 22

30

0

22

0

0

232

22

22

4

]DrurDrD

DrurD[

sinD

duDrurD

usinD

y)cosDrrD

dsincos(

sinD

y)dsinD(R

)cossinD(d )r(A

110

u, u

dsin)(cosdducosu let

57


Exp 11:Exp 11:

1

1

2222

22

1

1

22212222

1

1

232222

2122

1

1

221

1

22

1

1

1

1

1

1

22

22

1

1 22

23

2323

1

23

12

21

2

21

2

12

]DrurDrD

DrurD[

)]DrurD(Dru[)DrurD(rD

])DrurD(rD

)DrurD(Dr

u[

duDrurD )Dr

(DrurDDr

u

udv vuvdu

DrurDDr

vduDrurD

vd

duuduu

let, duDrurD

u

/

//

58


Exp 11:Exp 11:

rD, )r( r

D

rD, )r( D

rD, r

D

rD, D

rsinD

y

y)]rD)(DrrD(|rD|)DrrD)[(rD

(sinD

y]DrurDrD

DrurD[

sinD)r(A

3

40

0

2

230

222222

30

1

1

2222

2230

3

3

3

23

2

2

3

1

2

232

R

DD

y sinrr

59


Exp 11:Exp 11:

rD, ˆr

sin

D

rD, ˆsinr D

),,r(A

rD, )r( r

D

rD, )r( D

)r(A

2

40

0

3

40

0

3

3

3

3

DD

),,r(

60


Exp 11:Exp 11:

rD, ˆr

sin

D

rD, ˆsinr D

),,r(A

2

40

0

3

3

Dz D)ˆ sinr (cosD

ˆ]A)rA(r

[r

ˆ)]rA(r

Asin

[r

r]A)A(sin[sinr

AB

rr

000

3

2

3

2

3

2

111

1

For D r≧For D r≧

0

000

DD

),,r(

61

Summary Summary

62

Magnetostatic Boundary Conditions Magnetostatic Boundary Conditions

B

B

//B

63


belowabove

belowabove

BB

aBaB

adB

0

0

a

n

n

64


)nK(BB

KBB

KBB

IdB

belowabove

//below

//above

//below

//above

enc

0

0

0

0

n

65

66

67

68

69

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Magnetostatics