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```					Lecture 15 Magnetism: Magnetic materials
Introduction
If certain materials are introduced into the region
near a circuit then the self-inductance of that
circuit is found to change. This is similar to the
effect of a dielectric on a capacitor and the
treatment of magnetic materials follows very
closely the treatment of dielectrics covered in
Lectures 7 and 8.

In magnetic materials the observed effects are
due to the influence of the magnetic field on the
magnetic dipoles in the material.
Definition of relative permeability

The self-inductance of a circuit in vacuo is L0.

When all the region in which a magnetic field is
present is filled with a given material the self-
inductance changes to a new value Lm.

The relative permeability r of the material is defined
as

r=Lm/L0
Because the magnetic flux =LI this flux changes
as does the average magnetic field defined as B

r=m/0=Bm/B0

I=constant
Unlike r for dielectrics which always has a value >1
(r-1>0) some magnetic materials (diamagnetic ones)
have r-1<0 and some (ferromagnetic ones) have a r
which is highly non-linear (a function of the B-field)
and is dependent upon the previous history of the
material (exhibits hysteresis).

In the following treatment we will assume that our
magnetic materials are LIH (linear, isotropic and
homogeneous) ones although in practice this is often
a less reasonable assumption than for dielectrics.
Magnetisation and surface (Amperian)
currents

In a similar manner to the definition of polarisation
P for a dielectric we define a magnetisation M for
a magnetic material.

Each small volume d of a magnetised material
will possess a magnetic dipole moment dm. The
magnetisation is defined as the magnetic dipole
moment per unit volume

M=dm/d           The unit of M is Am-1
We can visualise each dipole in the material as
resulting from the flow of current around a small
loop. If all these loops (dipoles) are identical then
the currents at the interfaces between adjacent
elements cancel and only at the surface of the
material is there a net current.

I
I
The effects of the magnetic dipoles within the
material may be modelled by surface currents,
or surface current densities, known as
Amperian currents.

These Amperian currents are similar in effect
to the surface polarisation charges that were
introduced to explain the behaviour of
dielectrics.
I
I
Relationship between M and surface current
density Js

Consider a small element of the magnetised
material in the form of a cylinder of cross-section dS
and length dl. The magnetic dipole moment of this
element MdSdl follows from the above definition of
the magnetisation.
M
JS

dS

dl
The equivalent surface current density is Js so
that the total surface current is Jsdl.

This results in a magnetic dipole moment (=
current x area of circuit) of JsdldS.

As the two definitions of the dipole moment
must be equal we must have M=Js.
M
JS

dS

dl
If the surface is not parallel to M then this result is
slightly modified.

If M is not uniform then the currents on adjacent
loops within the material do not cancel and it is
necessary to also consider a volume Amperian
current.
B-fields in magnetic materials

A magnetic field B0 is produced in some region of
free space by an arbitrary conduction current or
currents.

A magnetic material is now introduced into this
region of space and becomes magnetised.

Within the material an additional field Bm is
produced which results from the existence of the
Amperian surface current Js.
The total B-field is now the sum of the original plus
the new field

B=B0+Bm
Ampère’s circuital law in the presence of
magnetic materials

In free space we have

   L
B  dL  0 I

This equation is still valid in the presence of
magnetic materials except that B is now the
total field and I must include both conduction IC
and Amperian IM currents

   L
B  dL  0 ( I c  I m )
After some manipulation

B       
 L  0  M   dL  Ic
        
this result gives the modified form of the circuital
law in the presence of magnetic materials.
The H-field

Because the quantity B/0-M occurs quite often it is
given a special name ‘magnetic field strength’ or H-
field, symbol H.
B
H        M   units Am-1
0

B  0 (H  M )
The circuital law for H is (from (A))

   L
H  dL  I c
from which the differential form may be derived
H=Jc
all currents (conduction and Amperian) may
contribute to B but only conduction currents may
contribute to H. H is the analogue of the
displacement field D in electrostatics.
In the absence of any magnetic materials M=0
and hence from (B) H=B/0. In any situation H
is given by the corresponding formula for B
divided by 0. For example for an infinitely long
wire

I
H
2 r

Because H can only arise from conduction
currents these equations are also valid in the
presence of magnetic materials.
Magnetic susceptibility and permeability
The magnetic susceptibility m at a given point is
defined as
M=mH
But B=0(H+M)= 0(1+m)H
Hence in the absence of magnetic materials
B=0H and in the presence of magnetic
materials B=0(1+m)H. As H remains constant
B must change by a factor (1+m) but this is also
the definition of r and so we have
r=1+m             (Definition of r)
Worked Example

An infinitely long solenoid has 100 turns per cm
and carries a current of 2A
Calculate the conduction surface current density Jc
and the field produced within the solenoid.
The solenoid is now filled with a material having
r=100. Calculate the B-field within the solenoid
and the Amperian surface current density Js.
What is the value of the magnetic field strength (H-
field) and magnetisation (M) for the above two
cases?
The equation B=0 is unmodified in the
presence of magnetic materials as there are still
no magnetic monopoles.

However because B=0(H+M)

 B=0=0(H+M)  H=-M

So sources of H are possible which must also
be sinks of M.
Boundary conditions for B and H

At a boundary between two different magnetic
materials there may be both a surface conduction
current Jc and an Amperian surface current Js.

B1        H1
H1t
1                                Jc
2                                Js
H2t
B2        dL
H2
Cylindrical Gaussian surface for the B-field.
Height of cylinder can be made infinitesimally
small  only the flux through the ends of the
B1   H1
cylinder need be considered.
H
Applying    B  dS  0           1
2
B1n-B2n=Bn=0                                    H2
B2      dL
Where B1n and B2n are the normal components
of the B-fields
Hence across any surface         the   normal
component of B is continuous.
For H we consider the loop of length dL and of
infinitesimal height

From    H  dL  I    c

(H1t-H2t)dL=HtdL=Ic  Ht=Ic/dL=Jc

The tangential component of H is discontinuous by
Jc (the conduction surface current density) across
any interface.          B1    H1
H1t
1                         Jc
2                         Js
H2t
B2    dL
H2
Magnetic energy in the presence of magnetic
materials
The magnetic energy stored by an inductor is
(1/2)LI2.
Consider a solenoid which is filled with a magnetic
material of relative permeability r. We have
L=A 0 rn2l and B=0rnI
Where A is the area of the solenoid, l is its length
and n is the number of turns per unit length.
Using the previous two equations to substitute
for L and I in the equation for the magnetic
energy

2
1 2 1                    B  1 B2
U  LI  ( A0 r n 2l )                       ( Al ) 
2    2                   0  r n    2 0  r
1
BH ( Al )
2

where the final term follows from B=0rH.
This result can be interpreted in terms of an
energy density multiplied by a volume. This
result for the energy density can be shown to
be a general one
1
Magnetic energy density = 2 BH
If B and H are not parallel then this result must
be written in the form
1
BH
2
These equations reduce to (1/2)0B2 in the
absence of magnetic materials.
In the presence of dielectrics we found that the
electrical energy density was given by

1         1
Electrical energy density =     DE or     DE
2         2
Conclusions
Definition of relative permeability (r)
LIH magnetic materials
Magnetisation (M) and surface (Amperian) currents
(Js) – relationship between these
B-fields in magnetic materials
Ampère’s circuital law in the presence of magnetic
materials
Magnetic field strength – H-field
Circuital law for H
Magnetic susceptibility and permeability r=1+m
Boundary conditions for B and H
Magnetic energy in the presence of magnetic
materials

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