0

Magnetization of a substance is its magnetic moment per unit volume

(similar to polarization in case of dielectrics in electric fields)

Total magnetic field at a point is a sum

total

V

M

M

B B 0

0 m 0All equations can be adapted by replacing K

Small magnetic effects are linear:

1

0 for diamagneticsMagnetic susceptibility

0 for paramagnetics

m mK

M

Magnetization

• Diamagnetism occurs in substances where magnetic moments inside atoms all cancel out, the net magnetic moment of the atom is zero. The induced magnetic moment is directed opposite to the applied field. Diamagnetism is weakly dependent on T.

• Diamagnetic (induced atomic moment) effect is overcome in paramagnetic materials, whose atoms have uncompensated magnetic moments. These moments align with the applied field to enhance the latter. Temperature T wants to destroy alignment, hence a strong (1/T) dependence.

Magnetic effects are a completely quantum-mechanical phenomenon, although some classical physics arguments can be made.

BM=C Curie's Law

T

Example: Magnetic dipoles in a paramagnetic material

Nitric oxide (NO) is a paramagnetic compound. Its molecules have maximum magneticmoment of ~ . In a magnetic field B=1.5 Tesla, compare the interaction energy of themagnetic moments with the field to the average translational kinetic energy of the moleculesat T=300 K.

23 5max

21

1.4 10 8.7 10

36.2 10 0.039

2

BU B J eV

K kT J eV

Ferromagnetism

Alignment of magnetic domains in applied field

• In ferromagnetic materials, in addition to atoms having uncompensated magnetic moments, these moments strongly interact between

themselves.

• Strongly nonlinear behavior with remnant

magnetization left when the applied field is lifted.

Permeability Km is much larger, ~1,000 to 100,000

Hysteresis and Permanent Magnets

Magnetization value depends on the “history” of applied magnetic field

Magnetization curve for soft iron showing

hysteresis

Example: A ferromagnetic materialA permanent magnet is made of a ferromagnetic material with a M~106 A/mThe magnet is in the shape of a cube of side 2 cm. Find magnetic dipole moment of a magnet. Estimate the magnetic field at a point 10 cm away on the axis

2

303

8

~ 10 102

total

total

MV A m

B T Gx

Experiments leading to Faraday’s Law

Electromagnetic Induction – Time-varying magnetic field creates electric field

Changing Magnetic Flux

No current in the electromagnet – B=0 - galvanometer shows no current.

When magnet is turned on – momentarily current appears as B increases.

When B reaches steady value – current disappears no matterhow strong B field is.

If we squeeze the coil as to change its area – current appearsbut only while we are deforming the coil.

If we rotate the coil, current appears but only while we arerotating it.

If we start displacing the coil out of the magnetic field – current appears while the coil is in motion.

If we decrease/increase the number of loops in the coil – current appears during winding/unwinding of the turns.

If we turn off the magnet – current appears while the magnetic field is being disappearing

The faster we carry out all those changes- the greater the current is.

Faraday’s Law quantified

effect theproduce will

flux magnetic changing Anything

cos

coil loop-Nan for

coil loop-single afor

BA

dt

dN

dt

d

B

B

B

( )0.24 0.048Bd d BA dB

A mV I mAdt dt dt R

Emf and Current Induced in a Loop

If the loop is made of the insulator, induced emf is still the sameBut the resistance is large, so little (or no current) is flowing

Circuit with induced EMF only

A1 B1 A2 B2

I1 – I3I1

I3

R1 R2R3

Kirchhoff’s rules still apply! It is only the origin

of the EMFs that is different here from ordinary batteries.

323121

21123

233231

13311

222

111

11

thatfollowsit e.g., And,

)(

:loopsfor equations Standard

Likewise,

EMF induced

yield field with Area

RRRRRR

RRI

RIRII

RIRI

dt

dBA

dt

dBA

BA

Direction of the induced EMF

Alternating current (ac) generators

tBA

tBABAB

sin

coscos

Direct current (dc) generators

Split ring (commutator) does the job of reversing polarity every half cycle

Motional emf – conductor moving in a constant magnetic field

€

FB = qvB will move charges

until compensated by the electric

field of end accumulations

qvB = qE = qV / l

V = Bvl

B Blx

dxBl Blvdt

2 2resistor

/

( ) /

I Blv R

P I R Blv R

Generators as Energy Converters

2

Who does the work?

We! - By moving the bar:

( ) /

Energy conserved

appliedP Fv IBlv Blv R Generator does not produce electric energyout of nowhere – it is supplied by whatever entity that keeps the rod moving. All it does is to convert it to a different form, namely toelectric energy (current)

20

:emf Total

:element Small

)(

:bar Rotating

2lBdrr

lB

drBvd

rrv

2

0

2

)/(

)/exp(

)(

:force magnetic

by the ddecelerate

relax illvelocity w

push, initialAfter

BlmR

tvv

vR

BlIBl

dt

dvm

Motion does not necessarily

mean changing magnetic flux!

Significance of the minus sign – Lenz’s Law

Induced current has such direction that its own flux opposes the change of the external

magnetic flux

Magnetic field of the induced current wants to decrease the total flux

Magnetic field of the induced current wants to increase the total flux

Correspondingly, magnetic forces oppose the motion – consistently with conservation of

energy!

Lenz’s Law – the direction of any magnetic induction effect as to oppose the cause of the effect

Lenz’s Law – a direct consequence of the energy conservation principle

Finding the direction of the induced current

Induced Electric Fields

sE

BE

BBv

BvF

F

BvEF

d

t

q

q

B

loop thearound integral line the

is which once, loop thearound chargeunit a

move todone work but the nothing is emf

! changingby induced field Electric

- then?charges drivesit that isWhat

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currents that show sexperiment sFaraday' BUT!

with )emf" motional("

conductors movingin currentsexplain did We

0 loop, in thecurrent have To

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