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).
Dr. Rakesh Choubisa, BITS, Pilani
Torque & Force on Magnetic Dipoles
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
0)( BldI
BldIF
).( BmF
Pr. 6.1: Calculate the torque exerted on the square loop due to the circular loop.
• Torque will be
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
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;
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
The Field of a Magnetized Object
Dr. Rakesh Choubisa, BITS, Pilani
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;
Dr. Rakesh Choubisa, BITS, Pilani
''
'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.
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
Dr. Rakesh Choubisa, BITS, Pilani
The Auxiliary Field H
• Ampere’s law in Magnetized Materials:• Total current through the material is (bound +
free currents);
• Using Ampere’s differential law;
Dr. Rakesh Choubisa, BITS, Pilani
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;
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
Boundary Conditions
Dr. Rakesh Choubisa, BITS, Pilani
),( belowabovebelowabove MMHH
nKHH fbelowabove ˆ)( ''''
Dr. Rakesh Choubisa, BITS, Pilani
Linear and Nonlinear Media
• When field is weak enough, we should write;
• But customary it is written in terms of H;
Dr. Rakesh Choubisa, BITS, Pilani
)!!!!(1
0IncorrectBM m
HM m
Magnetic susceptibility
Dr. Rakesh Choubisa, BITS, Pilani
Linear and Nonlinear Media
• Materials which obey this relation is called linear media
• Where;
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
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.
Dr. Rakesh Choubisa, BITS, Pilani
z x
y