Download Magnetic properties of materials- I

Magnetic
properties of
materials- I
Magnetic properties
of materials- I
Magnetic dipoles
Field of a magnetic dipole
 Force on a dipole in a non uniform
field
 Induced magnetic dipole moment
Circular current loop
B
p
dB
B
r
z


0 I
2
R
R
2
2
z

2 3/ 2
Circular current loop
B
0 I
2
R
R
2
z
If z >>R
B
0 I R
2 z
2
2
3
 0 I R
B
3
2 z
2

2 3/ 2

0 
B

3
2z
Where  is =
2
R
I
Known as Magnetic dipole moment
•  and B are in the same direction.
• B is produced by .
• In general,  = I A,
true for any shape of
the loop.
• A is a vector area,
with direction assign
by right hand rule.
Magnetic dipole in an external
Magnetic field
Magnetic
dipole
experiences a
torque
Torque of a current loop
B  B0 zˆ
z
n
h
y
e
x
g
f
Torque on a current loop
z
B  B0 zˆ
n
h
y
e
g
a
b
x
f
Plane of the loop makes
angle  with the field
n̂

B
z

e
b
y

f
Fef  IbB sin( 90   ) ( xˆ )
n̂

B
z

h
b
y

g
Fgh  IbB sin( 90   ) ( xˆ )
z
n
Forces on a current loop

h
e
y
g
b
x
a
f
F fg  IaB
Fhe  IaB
( y direction )
( y direction )
y
Forces on a current loop
Fef  IaB sin( 90   )
Fgh  IaB sin( 90   )
( x direction )
( x direction )
F fg  IaB
Fhe  IaB
( y direction )
( y direction )
Contribute to torque
 
  r F
n̂
b
 R  IaB sin  xˆ
2
b
 L  IaB sin  xˆ
2
z
Fhe

e

y


f
Ffg
Torque on a current loop
  IabB sin  xˆ
  IAB sin  xˆ

  NIA (nˆ  B)
Torque of a magnetic
dipole


  IA nˆ  B
  
  B
Torque tends to rotate
 so that it lines up
with B.
Potential energy for the
dipole
• B makes an angle  with the dipole
U    d


U    B
U has smallest value when  and B are
parallel 
- B
 Largest when anti-parallel  B
Find the magnetic dipole moment
of the circuit.
  iA
 i
 a  b
2
2
2

Find the magnetic
dipole moment of the
loop. All sides have
equal length and it
carries a current I.
z
a
a y

x
a
  Ia yˆ  Ia zˆ
2
2
Direction is along
the line y = z.
Important to note
• In general, potential energy (PE) can not
be defined for a Magnetic field alone.
• Since torque on the dipole depends upon
the its position with respect to the field,
PE can be defined for magnetic dipole in
the field.
• This PE corresponds to any change in
the rotational configuration.
Comparison with electric
dipole

P
U = -  B cos
-PE cos
Monopole does
not exist
Independent of
choice of origin
=xB
Single charge
exists
Only when total
charge vanishes.
=PxE
The field of a
magnetic dipole
Electric field of an electric
dipole
Magnetic field lines of a
magnetic dipole
Bar magnet can also be
considered to be a magnetic
dipole.
Field lines do
not start or end
but
continue
through
the
interior of the
magnets,
forming close
loops.
Similarities
Electric and magnetic dipole fields vary as r-3
when we are far from the dipoles.
Force on a dipole in a
nonuniform field
In a uniform field total force on the dipole
(electric as well as magnetic) is zero.
There is only torque but no net motion
In a non uniform field net force is not
zero. Dipole may move.

 
dF  i2 ds  B1
Net force
will be
downward
on the loop.
 
U 2    2B1
U 2   2 z B1z
U 2   2 z B1z
dU
F21  
dz
d
F21     2 z B1z 
dz
dB1z
F21   2 z
dz
If,
2z  0
dB1z
0
dz
F21< 0
Force on the loop 2 due to 1 is
attractive (downward).
Induced magnetic
dipole moments
Induced magnetic dipole
moments
• An
applied
magnetic
field can
induce
dipole
moments
I-Iind
I+Iind
Downward
Induced magnetic
dipole moment.
Important points
• In a non uniform magnetic field,
permanent dipoles are attracted
towards the source of the field
• Induced dipoles are repelled from the
source of the field.
• Similar effect is observed in materials
that lack permanent magnetic dipole
moments.
An idea about magnetic
moments
System
Nucleus
Magnetic dipole
Moment (J/T)
~ 10-28
Electron
~ 10-23
Bar Magnet
5
Earth
1022
Atomic and nuclear
magnetism
• Magnetic properties depend upon
magnetic properties of the individual
atoms.
• Magnetic material is consists of atomic
dipoles
dipole moment
associated with circulation of electron.
Calculation
• We consider magnetic materials to be
composed of a collection of atomic
dipoles.
• These dipoles might align when an
external electric field is applied.
• An electron circulating about the
nucleus can be considered as a current
loop of radius r and speed v.
Calculation
• Current in the loop =
q qv
i 
T 2r
ev 2 evr
  iA 
r 
2r
2
• Bohr’s model
• Orbital magnetic dipole moment
nh
mvr 
2
el
l 
2m
Bohr’s magneton
• This is a basic unit of atomic
magnetic dipole moment
ev 2 evr
  iA 
r 
2r
2
nh
mvr 
2
eh
 24
B 
 9.27 10 J / T
4m
Nuclear magnetism
Nuclear
magnetic
moments
Orbital part
Intrinsic part
(spin)