chapter-6 : magnetic fields in matter magnetization – dia-magnets, paramagnets, ferro-magnets –...

Chapter-6 : Magnetic Fields in Matter

• Magnetization– Dia-magnets, Paramagnets, Ferro-magnets – At microscopic level, electrons in atom

revolve around the nucleus as well have spin. This will form tiny current loop and hence dipole moment.

– Ordinarily the dipole moments in matter cancel each other due to random orientation of atoms.

Dr. Rakesh Choubisa, BITS, Pilani

What happens when a material is subjected to external magnetic

field.• Unlike electric field, magnetic dipoles aligns

either along field (para-magnetism) or opposite to it (Diamagnetism) .

• Some materials retain magnetic property even after switching off the magnetic field (Ferromagnetism).


Torque & Force on Magnetic Dipoles


θm

I

Y

X

Z

θ

F

F

Y

Z

abImwhereBmN

xaFN

;

ˆ)sin(

mθ

θa

b

Torque & Force on Magnetic Dipoles

• In a uniform magnetic field; net force is zero.

• While in non-uniform magnetic field, it is non-zero and can be expressed as (Pr. 6.3);

Similar to the electrostatics case.


0)( BldI

BldIF

).( BmF

Pr. 6.1: Calculate the torque exerted on the square loop due to the circular loop.

• Torque will be


xr

baIN ˆ

4 3

2220

Effect of a magnetic field on atomic orbit

• Electrons in atom revolves around nucleus and contribute steady current of period T.


Z

m

e-

Rv

B T

eI

zRve

rIm ˆ2

)( 2

Effect of a magnetic field on atomic orbit

• Electron will speed up by the amount;

• Hence change in magnetic dipole;


em

RBev2

em

BRez

Rvem

4ˆ

2

222

Dipole always aligns opposite to filed

Magnetization

• In the presence of external magnetic field;• We have a net polarization either

• Along the field (Para-magnetism related to spin of unpaired electrons)

• Opposite to the field (Dia-magnetism related to the orbital motion of electrons)

• For this we define magnetization as magnetic dipoles per unit volume.


The Field of a Magnetized Object


m

rs

dΓ’

'2

'0 ˆ)(

4)(

dr

rrMrA

s

s

'''0 )]ˆ

1()([

4)(

dr

rMrAs

The Field of a Magnetized Object

• Using product rule 7, we have;


''

'0'''0 )(

4)(

1

4)(

dr

rMdrM

rrA

ss

')(1

4)(

1

4)( '0'''0 adrM

rdrM

rrA

ss

Volume bound current JbSurface bound current Kb

Problem 6.8 A long cylinder of radius R carries a magnetization

Find the magnetic field.


2skM

MskB

zRknMK

zskMJ

b

b

02

0

2

ˆ

ˆˆ

ˆ3

Physical interpretation of bound currents

• When we have uniform magnetization, we have, on average, surface bound current due to cancellation of currents of all the internal sides of tiny current loops within the material.

• In a non-uniform magnetization, we have also volume bound current due to net current from the internal tiny current loops.

• In both cases; current is due to motion of bound charges attached to each atom.



The Auxiliary Field H

• Ampere’s law in Magnetized Materials:• Total current through the material is (bound +

free currents);

• Using Ampere’s differential law;


fb JJJ

ffb JMJJB

0

1

The Auxiliary Field H

• We get differential form of Ampere’s law for H

• Also, we have the integral form;


fJHMB

0

encfIldH

Pr. 6.12: An infinite long cylinder, of radius R, carries a frozen-in magnetization, parallel to the axis, , and there is no free current anywhere. Find the magnetic field.

• It can also be solved using H.Dr. Rakesh Choubisa, BITS, Pilani

zskM ˆ

)(0);(ˆ

ˆˆ

ˆ

0 outsideInsidezskB

RknMK

kMJ

b

b

A Deceptive Parallel

• H and B are not like even if we have similar from of Ampere’s law.

• As field is found by both curl and div. of vector, div. of B is always zero however div. of H in general is not zero (as, div. M is not zero) & hence both are not like.

• However, when Div. of M is zero, the parallel between B and µ0 H is faithful.


Boundary Conditions


),( belowabovebelowabove MMHH

nKHH fbelowabove ˆ)( ''''


Linear and Nonlinear Media

• When field is weak enough, we should write;

• But customary it is written in terms of H;


)!!!!(1

0IncorrectBM m

HM m

Magnetic susceptibility


Linear and Nonlinear Media

• Materials which obey this relation is called linear media

• Where;


HMHB m)1()( 00

HB

)1(0 m

Permeability of the material

Pr. 6.17: A current I flows down a long straight wire, made of linear material with susceptibility , of radius a. Find the magnetic field, all bound currents and the net bound current flowing down the wire. The current I is distributed uniformly.


m

0

)(2

ˆ

)(

)(2

);(2

)1(

2

02

0

currentTotal

Itooppositea

InMK

Ialonga

IMJ

outsides

IInside

a

IsB

mb

mb

m

Pr. 6.25: A toy consists of donut-shaped permanent magnet. (a) If you put two back-to-back magnets on the rod. At what height (z) does the upper one float?(b) If you now add a third magnet, write the equation of motion for the floating condition for the upper magnets.


z x

y

chapter-6 : magnetic fields in matter magnetization – dia-magnets, paramagnets, ferro-magnets –...

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