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Magnetization
Magnetization of a substance M is its magnetic moment per unit volume
(similar to polarization in case of dielectrics in electric fields)

M

 total
V
Total magnetic field at a point is a sum
B  B 0  0M
All equations can be adapted by replacing 0  K m 0
Small magnetic effects are linear:
m  Km  1
   0 for diamagnetics
Magnetic susceptibility 
   0 for paramagnetics
• 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.
B
M=C   Curie's Law
T
Magnetic effects are a completely quantum-mechanical phenomenon,
although some classical physics arguments can be made.
Example: Magnetic dipoles in a paramagnetic material
Nitric oxide (NO) is a paramagnetic compound. Its molecules have maximum magnetic
moment of ~ B . In a magnetic field B=1.5 Tesla, compare the interaction energy of the
magnetic moments with the field to the average translational kinetic energy of the molecules
at T=300 K.
U max   B B  1.4  1023 J  8.7  105 eV
3
K  kT  6.2  1021 J  0.039 eV
2
Ferromagnetism
• 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
Alignment of magnetic
domains in applied field
Hysteresis and Permanent Magnets
Magnetization value depends on the “history” of applied magnetic field
Example: A ferromagnetic material
A permanent magnet is made of a ferromagnetic material with a M~10 6 A/m
The 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
total  MV  8 A  m 2

3
B ~ 0 total

10
T  10 G
3
2 x
Magnetization curve for
soft iron showing
hysteresis
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 matter
how strong B field is.
If we squeeze the coil as to change its area – current appears
but only while we are deforming the coil.
If we rotate the coil, current appears but only while we are
rotating 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


d B

for a single - loop coil
dt
d B
 N
for an N - loop coil
dt
 B  BA cos 
Anything changing magnetic flux
will produce the effect
Emf and Current Induced in a Loop
d  B d ( BA) dB


A  0.24mV
dt
dt
dt
I

R
 0.048mA
If the loop is made of the insulator, induced emf is still the same
But the resistance is large, so little (or no current) is flowing
Circuit with induced EMF only
Area A1 with field B1 yield
A1 B1
R1
A2 B2
R3
R2
I3
I1
I1 – I3
Kirchhoff’s rules still
apply! It is only the origin
of the EMFs that is
different here from
ordinary batteries.

dB1
dt
dB2
Likewise,  2   A2
dt
Standard equations for loops :
induced EMF
1   A1
I1 R1  I 3 R3   1
( I1  I 3 ) R2  I 3 R3   2
And, e.g., it follows that
R2 1  R1 2
I3 
R1 R2  R1 R3  R2 R3
Direction of the induced EMF
Alternating current (ac) generators
 B  BA cos   BA cos t
  BA sin t
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
 B  Blx
FB  qvB will move charges
until compensated by the electric
field of end accumulations
qvB  qE  qV /l
V  Bvl

dx
  Bl
  Blv
dt
Generators as Energy Converters
I  Blv / R
Presistor  I 2 R  ( Blv )2 / R
Generator does not produce electric energy
out 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 to
electric energy (current)
Who does the work?
We! - By moving the bar:
Papplied  Fv  IBlv  ( Blv )2 / R
Energy conserved
After initial push,
Rotating bar :
velocity w ill relax
v(r )  r
Small element :
decelerate d by the
magnetic force :
  Bv dr
d
Total emf :

l
Bl 2
  Br dr 
2
0
2
m
dv
( Bl )
  IBl  
v
dt
R
v  v0 exp( t /  )
  mR /( Bl ) 2
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
No matter wha t , the total force on a charge is
F  q (E  v  B )
To have current in the loop, F  0
We did explain currents in moving conductors
(" motional emf" ) with FB  qv  B
BUT! Faraday' s experiment s show that currents
are induced when v  0 but B  B(t )
What is it that drives charges then? Electric field E induced by changing B !

emf
is nothing but the work done to move
a unit charge around the loop once, which is
the line integral around the loop
 E  ds