7 magnetostatic

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Magnetic MaterialsMagnetic Materials

www.phy.iitb.ac.in/~ph102www.phy.iitb.ac.in/~ph102

04/10/2304/10/23 22

Magnetic MaterialMagnetic Material

Magnetic field in materials is due to tiny electric Magnetic field in materials is due to tiny electric current loops or current loops or magnetic dipolesmagnetic dipoles. At normal . At normal temperatures the magnetic moments of the temperatures the magnetic moments of the dipoles are randomly oriented so that the net dipoles are randomly oriented so that the net magnetic moment is zero.magnetic moment is zero.Magnetization = Magnetic moment per unit Magnetization = Magnetic moment per unit volumevolumeWhen an external magnetic field is applied the When an external magnetic field is applied the dipoles align and the material develops a dipoles align and the material develops a magnetization.magnetization.

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Dia-, Para-, and FerronmagnetismDia-, Para-, and Ferronmagnetism

Electric dipoles always align in the direction of Electric dipoles always align in the direction of magnetic field but different magnetic material magnetic field but different magnetic material behave differently. behave differently.

Diamagnetic material have their manetization Diamagnetic material have their manetization direction opposite to that of B.direction opposite to that of B.

Paramagnetic material have M parallel to BParamagnetic material have M parallel to B

Ferromagnets are those which retain their Ferromagnets are those which retain their magnetization even after the magnetic field is magnetization even after the magnetic field is withdrawn – Hysteresis. withdrawn – Hysteresis.

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Types of magnetic materialsTypes of magnetic materials

ty.permeabili relative

theisμ andty permeabili absolute theisμ Here

HHH)1()H(B

ic.paramagnet isit positive, isit If

c.diamagneti is material thenegative, is If

litysusceptibimagnetictheiswhere HM

r

r0m0m0

m

mm

H

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Paramagnetic material in Paramagnetic material in MagneticMagnetic Field Field

Atomic magnets are oriented in direction Atomic magnets are oriented in direction of B, currents in adjacent loops cancel of B, currents in adjacent loops cancel giving rise to a surface currentgiving rise to a surface current

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Force on a magnetic dipoleForce on a magnetic dipole

Force on a current carrying loop isForce on a current carrying loop is

A

B

zero. is loopcurrent closed aon force theHence

)( BldIF

L

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A current loop experiences a force in an A current loop experiences a force in an inhomogeneous Fieldinhomogeneous Field

x

y

z

O A

BC

0 ,,0

00

ˆ

),,0(ˆ)0,,0(ˆ

BC&OA sidesOn

dyz

BjI

dyyBjIdyyBjIF

y

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Dipole in Inhomogeneous Magnetic FieldDipole in Inhomogeneous Magnetic Field

x

y

z

O A

BC

0 ,0,0

00

ˆ

),0,0(ˆ),,0(ˆ

OC&AB sidesOn

dzy

BkI

dzzBkIdzzBkIF

z

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Dipole in Inhomogeneous Magnetic FieldDipole in Inhomogeneous Magnetic Field

Assume that the derivatives are constant at the Assume that the derivatives are constant at the boundaries of small loopsboundaries of small loops

iIεmB

Bmz

Bk

y

Bj

y

BiI

z

Bi

z

Bk

y

Bi

y

BjI

dzz

Bjdz

y

BkIF

xxx

zxyx

yz

ˆ and 0 as

)(ˆˆˆ

)ˆˆ()ˆˆ(

)ˆ()ˆ(

2

2

2

0 0 ,,0,0,0

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Torque on a current loopTorque on a current loop

A dipole experiences a torque in a constant magnetic A dipole experiences a torque in a constant magnetic fieldfield

Torque is zero when the magnetic moment is parallel to Torque is zero when the magnetic moment is parallel to the field. To bring the dipole to a position which makes the field. To bring the dipole to a position which makes an angle an angle with the magnetic field, one has to do work. with the magnetic field, one has to do work.

Bm

Bm-U

mB

dmBdW

dipole ofEnergy Potential

)cos1(

sin0

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Field due to a magnetic dipoleField due to a magnetic dipole

For paramagnetic material, magnetization is For paramagnetic material, magnetization is proportional to the external field.proportional to the external field.

Magnetic field on the axis of a circular coil of radius Magnetic field on the axis of a circular coil of radius a was seen to bea was seen to be

! dipole electric

an for field electric of that similar tovery

)(for 2)(2 3

02/322

20 az

z

m

za

aI

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Vector Potential of a current loopVector Potential of a current loop

r

ldIA

4

0

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x

y

z

P

O

dl

r

dA

x

O torelativeposition r

ld from P of distance

plane z-in xpoint field

planey -in x is loopcurrent

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x

y

z

P

O

dl

r

dA

x

dl’

AdAd

Al|| dAd

ofcomponent -y addsbut

ofcomponent - xcancelshich element w

oppositelly symmetrica a ,ldeach For

component. -z no has , since

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dr

a

r

a

r

a

rak

I

dr

a

r

a

rak

I

darar

ak

I

da

kIldI

A

)2

cos3cos

21(

1cosˆ

4

)cos2

1(1

cosˆ4

)cos2(

cosˆ4

cosˆ44

2

22

2

22

0

0

2/12

22

0

0

2

02/122

0

2

0

00

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rmr

kr

mk

r

mx

r

xk

Ia

dr

axak

I

dr

xa

r

ax

r

a

rak

IA

xr

30

20

30

3

20

2

03

20

4

222

22

22

0

0

4

ˆ4

sinˆ4

ˆ4

cosˆ4

)2

cos3cos

21(

1cosˆ

4

coscos Use

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Vector Potential of magnetizedVector Potential of magnetized substance substance

If M is the magnetization, a volume dIf M is the magnetization, a volume dcontains contains MdMddipoles. The vector potential at a point P is dipoles. The vector potential at a point P is given bygiven by

drr

M

drr

rrMrA

1

4

)(

4)(

0

30

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Vector Potential of magnetizedVector Potential of magnetized substance substance

SdMrr

dMrr

drr

MdMrr

rA

rrMM

rrrr

M

11

4

11

4)(

11

Use

0

0

sdd

surfacevol

vv

used have westep,last In the

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Bound currentsBound currents

nMrK

MrJ

dSrr

rKd

rr

rJ

SdMrr

dMrr

b

b

surface

b

vol

b

ˆ)(

)(

)(

4

)(

4

11

4)rA(

00

0

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Uniformly magnetized slab Uniformly magnetized slab

t

K

M

nMrK

MrJ

b

b

ˆ)(

)(

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Free and Bound CuFree and Bound Currrentsrents

Net current is due to actual transport of Net current is due to actual transport of charges while bound current is due to charges while bound current is due to magnetization.magnetization.

Ampere’s Law for magnetized material Ampere’s Law for magnetized material contains contribution due to both types of contains contribution due to both types of current and is,current and is,

boundfree JJJ

)(

)(

0

00

MJ

JJJB

f

bf

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Fields B, H and MFields B, H and M

Define H-field byDefine H-field by

MH

I

JH

enclosedfree

f

Also

ldH

ly,Equivalent

)(

MB

H

0

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ExampleExample

An infinitely long cylinder of radius R contains a An infinitely long cylinder of radius R contains a frozen in magnetization M=kr, where r is distance frozen in magnetization M=kr, where r is distance from axis and no free current. Find B, H, M.from axis and no free current. Find B, H, M.

ˆ|ˆˆ|ˆ

ˆ

ˆˆ1

current volumeBound

ˆ

kRrkkrnMK

k

r

Mr

M

rMJ

kkrM

RrRrb

zzb

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Infinitely long cylinderInfinitely long cylinder

Use Ampere’s Law. As the current is Use Ampere’s Law. As the current is inindirection, field are along the axis. B is zero direction, field are along the axis. B is zero outside the cylinder.outside the cylinder.

rL

kkrB

kLrrRLkkLR

LdrJLKBLldBR

r

bb

ˆ

))()((

)(

0

00

0

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Magnetized cylinder (contd.)Magnetized cylinder (contd.)

k̂

cylinder outside 0MB

M-B

0H0

alone. curlby specified is M , 0

0

0

0

kr

ldH

z

MM z

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A uniformly magnetized CylinderA uniformly magnetized Cylinder

sinˆ

0

MnMK

MJ

b

b

n

M

z

R

Rsin

rld ˆ

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Uniformly magnetized sphereUniformly magnetized sphere

From symmetrically located pairs of elements horizontal From symmetrically located pairs of elements horizontal components cancel while vertical components add up.components cancel while vertical components add up.

n

M

z

R

Rsin

rld ˆ

M

kMdkMB

kdMkRR

M

kdldR

MR

dMRRdKdI

kR

dldI

r

rlddIBd

b

0

0

0

30

302

0

2

20

ring

20

20

3

2

3

4ˆ2

sinˆ2

ˆsin2

ˆsin2sin

2

ˆ sin

2dB

ring thefrom dB on tocontributi

sin

ˆsin

2

ˆ

4

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Example : Large Piece of uniformly magnetized Example : Large Piece of uniformly magnetized material from which a sphere is scooped out.material from which a sphere is scooped out.

B0

MH

MBB

H

M

MB

H

MBB

M

MB

H

3

1

3

2

,0 hole theof centre at the Since

3

2

ion magnetizat

uniform of sphere a with void theFill

oid, without vmaterial For the

0

0

0

0

0

0

00

0

00

7 magnetostatic

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