Magnetic Materials - Electricity and Magnetism - Lecture Notes, Study notes for Electricity and Magnetism. Dhirubhai Ambani Institute of Information and Communication Technology

Electricity and Magnetism

Description: This is the Lecture Notes of Electricity and Magnetism which includes Vector Potential, Boundary Conditions, Vector Derivative Operator, Vector Calculus, Vector Calculus, Three Co-Ordinate Systems, Two Types of Charge, Coulomb's Law, Electric Field etc. Key important points are: Magnetic Materials, Atomic Origins of Magnetic Properties, Maxwell's Equations, Electric Dipole Density, Magnetic Moment, Fundamental Quantum, Angular Momentum of Electron
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PHY481 - Lecture 29: Magnetic materials
Griffiths: Chapter 6
Magnetic materials
Extension of Maxwell’s equations to treat magnetic fields inside and outside magnetic materials is achieved in
manner that in some ways is like the treatment of the dielectric response of materials - through analysis of dipoles, in
the magnetic case of course we consider magnetic dipoles. To treat a large number of aligned electric dipoles inside
a material using Maxwell’s equations, we introduced the polarization ~
P, the electric dipole density. To treat a large
number of aligned magnetic dipoles using Maxwell’s equations we introduce an analogous quantity, the magnetization
~
Mwhich is the magnetic moment density. In the dielectric case, atoms and molecules have either an induced or
intrinsic electric dipole moment. In a similar way, atoms and molecules may have an intrinsic magnetic moment
and/or a magnetic moment induced by an applied magnetic field.
Atomic origins of magnetic properties
The magnetic properties of atoms comes from a combination of the orbital motion of the electrons about the
nucleus and the intrinsic magnetic moment of electrons, protons and neutrons. The orbital response produces a
magnetic moment that opposes an applied magnetic field, while the spin part typically produces a magnetic moment
that enhances an applied magnetic field.
The fundamental quantum of electron magnetic moment - the Bohr magneton
If a single electron moves in a circular orbit at speed vand with radius r, then the current is
I=e
2πr v(1)
The magnetic moment, morbital is then,
morbital =Iπr2=1
2evr =e
2me
L(2)
where L=mevr is the angular momentum of the electron. This relation is usually written as,
~morbital =gL~
L(3)
where gL=e
2meis called the gyromagnetic ratio.
According to Bohr theory, the angular momentum is quantized, so that,
L=l¯h(4)
where l= 0,1,2... is an integer. We then have,
morbital =e
2me
¯hl =mBl(5)
where we have defined the Bohr magneton,
mB=e¯h
2me
= 9.27 ×1024Am2(6)
In addition, the electron carries an intrinsic magnetic moment due to its spin. The intrinsic magnetic moment of the
electron has a value very close to mB.
The total magnetic moment is found by a vector addition of the orbital and spin contributions. This sum is called
~m0. The magnetization used in Maxwell’s equations are the density of aligned moments ~m0.
Response of electron spins and electron orbits to an applied magnetic field
Above we showed that an electron in circular orbit may have orbital or spin contributions, however in many
situations neither of these contributions is large. The orbital contributions may be small in s-states or if there are
electrons in each of the possible high angular momentum states, such as p-states. Moreover, in most situations, the
response of the electron spin to an applied field depends on whether the atom or molecule has paired or unpaired
electrons in each orbital. If the electrons are paired, the spin response is weak as the paired electrons must be broken
up in order for the spins to align with an applied field. In contrast if there are unpaired electrons in an atom or
2
molecule, the unpaired electrons can easily line up with an applied field. Paramagnetism and ferromagnetism occur
when electron spins line up in a common direction. In the case of paramagnetism, an applied field is used to line up
the spins. In ferromagnetic materials the spins can remain lined up even when an applied field is switched off.
Orbital contributions to the magnetic moment are almost always zero until the atom or molecule is placed in an
applied field. The qualitative effect is that an applied magnetic field produces a diagmagnetic effect where the induced
dipole moment is opposite to the original direction of the dipole in a circular orbit. This can be seen by considering
the relationship between radius and velocity for the Bohr atom, ie
mv2
r=ke2
r2; or vr =ke2
mv (7)
Now consider applying a small magnetic field that does not significantly alter this relationship, though of course a
magnetic force q~v ~
Bdoes occur. The orbital moment is given by evr/2 so increasing the product vr leads to an
increase in the orbital moment. As seen from the equation above, smaller velocities (larger orbits) leads to larger
orbital magnetic moment of course this is consistent with Bohr theory as vr is proportional to the magnitude of the
angular momentum.
However the most strongest diagmagnetism occurs in superconductors where the effect can be perfect so as to screen
out an applied field completely. In these materials superconducting screening currents are set up to completely cancel
an applied magnetic field. In that sense they are qualitatively similar to the perfect dielectric behavior of metals
where static charges perfectly screen out an applied electric field. As we shall see below there are some important
differences that are not evident in these qualitative analogies.
The bottom line is that most atoms and molecules that have unpaired electrons are likely to be dominated by
paramagnetic response, while those that have an even number of electrons, assuming that the electrons are paired,
are typically diamagnetic. Superconductors are a special case where diamagnetism can be very strong.
Maxwell’s equations for magnetic materials
We would like to extend Maxwell’s equations to treat the magnetic fields inside and outside magnetic materials
so we would like to replace the magnetization with current sources. It turns out that the relations between bound
currents and the magnetization are,
~
Kb=~
M~n;~
jb=~
∇ ∧ ~
M(8)
If the magnetization inside a material is a constant, only surface currents are required to produce the magnetization,
while if the magnetization varies in space, a bulk bound current density is also needed. The physical reasoning leading
to these observations is constructed by taking local current loops and showing that if the local current loop density
is a constant only a surface current remains. The formal calculation of these results is as follows.
The vector potential due to a magnetized material is found by adding up the vector potential contribution from
each part of the material, so that,
~
A(~r) = µ0
4πZ~
M(~r0)(~r ~r0)
|~r ~r0|3 0=µ0
4πZ~
M(~r0)~
0(1
|~r ~r0|) 0(9)
Product rule 7 of Griffiths,
~
(f~
A) = f(~
∇ ∧ ~
A)~
A~
f(10)
leads to,
~
A(~r) = µ0
4πZ(1
|~r ~r0|~
0~
M(~r0)) 0µ0
4πZ~
0(~
M(~r0)
|~r ~r0|) 0(11)
Using the vector identity (see Problem 1.60b of Griffiths)
Z(~
∇ ∧ ~
A)0=I~
Ad~a (12)
we finally have,
~
A(~r) = µ0
4πZ(1
|~r ~r0|~
0~
M(~r0)) 0+µ0
4πI1
|~r ~r0|(~
M(~r0)ˆn0)da0(13)
3
The first term is a superposition formula for bulk currents, provided ~
jb=~
~
M, while the second integral corresponds
to a superposition of surface currents provided we define ~
Kb=~
Mˆn.
Incorporating bound currents into Ampere’s law
If we have a problem where we know the magnetization, we can solve for the magnetic fields using Ampere’s law
where,
~
∇ ∧ ~
B=µ0(~
jf+~
jb) = µ0(~
jf+~
∇ ∧ ~
M) = µ0(~
∇ ∧ ~
H+~
∇ ∧ ~
M) (14)
where we defined ~
Hto be the magnetic intensity and it obeys,
~
∇ ∧ ~
H=~
jfso that I~
H·d~
l=if.(15)
From this equation it is seen that Amperian contours can be used to relate ifto the magnetic intensity (or Auxilliary
field) ~
H. The magnetic intensity and magnetization have units Amp per meter (A/m) as can be seen from the
definition of ~
H. Once the magnetic intensity has been found, the magnetic field is simply,
~
B=µ0(~
H+~
M) (16)
A uniformly magnetized cylinder with length >> radius
We take the magnetization to be uniform, ~
M=M0ˆzand to be oriented along the axis of the cylinder. There are no
free currents, so there is no gain in using the magnetic intensity (H= 0 as if= 0) and ~
B=µ0~
M. The magnetization
is uniform, so the curl of the magnetization is zero, therefore the only bound currents are the surface currents ~
Kb.
Since the magnetic field and magnetization are uniform, we can think of the problem as being similar to the the
uniform field inside a solenoid, so we expect to find bound surface currents flowing around the cylinder. We find the
suface currents through,
~
Kb=M0ˆzˆs=M0ˆ
φ(17)
This result applies in central regions of the magnetized cylinder. If we now consider a point ~r which is well removed
from the cylinder, i.e. r >> l, then we can treat the cylinder as a dipole with dipole moment ~mcyl =V olume ×~
M,
and then find the field using the dipole formula,
~
B(~r) = µ0
4π
3(~mcyl ·ˆr)ˆr~m
r3(18)
As a check, lets use Ampere’s law to find the magnetic field inside the cylinder. An amperian loop leads to
Bl =µ0Kbl, so that ~
B=µ0Kb.
A Uniformly magnetized sphere of radius R
Take the magnetization to be, ~
M=M0ˆz, so that the bound currents are now,
~
Kb=~
Mˆr=M0sinθ ˆ
φ(19)
Assume that the magnetic field inside is uniform and along the ˆzdirection, and that outside it is like a dipole, so
that,
~
Bint =B0ˆz; and ~
Bext =µ0
4π
3(~msphere ·ˆr)ˆr~m
r3(20)
The boundary conditions on the magnetic field at the sphere surface are, Bext
tBint
t=µ0Kb,Bext
n=Bint
n. The
latter equation gives,
B0ˆz·ˆr=B0cosθ =µ0
4π
2~msphere ·ˆr)
R3=µ0
4π
2mspherecosθ
R3(21)
From this expression, we find that
B0=µ0
4π
2M04πR3
3R3=2µ0M0
3(22)
4
and hence,
~
Binside =2µ0M0
3ˆz(23)
It is easy to show that the boundary condition on the tangential component is also satisfied, so we have the correct
solution. Notice that this solution differs by a factor of two from the analagous solution for a uniformly polarization
sphere.
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