15 and 16 Magnetism: Magnetic materials
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
8 and 9.
In magnetic materials the observed effects are due to the influence
of the magnetic field on the magnetic dipoles in the material.
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 mr of the material is defined as
Because the magnetic flux F=LI
this flux changes as does the average magnetic field defined as <B>
Unlike er for dielectrics which always has a value >1 (er-1>0)
some magnetic materials (diamagnetic ones) have mr-1<0 and some (ferromagnetic ones) have a mr
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.
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 dt of a magnetised material will possess a
magnetic dipole moment dm. The
magnetisation is defined as the magnetic dipole moment per unit volume
The unit of M is Am-1
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.
The effects of the magnetic dipoles within the material may be
modelled by surface currents, or surface current densities, known as Amperian
These Amperian currents are similar in effect to the surface
polarisation charges that were introduced to explain the behaviour of
between M and surface current density Js
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.
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.
If the surface is not parallel to M then this result is slightly
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
in magnetic materials
A magnetic field B0
is produced in some region of free space by an arbitrary conduction current or
A magnetic material is now introduced into this region of space and
Outside of 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
circuital law in the presence of magnetic materials
In free space we have
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
now apply this result to a general path which includes conduction currents and
magnetic materials. In the figure the path L
links only the conduction current IB
and the magnetic material B.
In order to
evaluate the circuital law in this case we need to calculate the contribution to
IM as the path passes
through material B.
This can be found by referring to the following diagram. L
makes an angle a
to M and therefore has a component dl=dLcosa
In terms of the surface current density Js
the current linked by the portion of the path dL as it passes through the material is (current = surface current density x length)
using the fact that M=Js
Hence the circuital law becomes
or combining the two line integrals
this result gives the modified form of the circuital law in the
presence of magnetic materials.
Because the quantity B/m0-M
occurs quite often it is given a special name ‘magnetic field strength’ or H-field,
from which the differential form may be derived
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/m0.
In any situation H is given by the
corresponding formula for B divided
For example for an infinitely long wire
Because H can only arise
from conduction currents these equations are also valid in the presence of
The equation Ñ×B=0
is unmodified in the presence of magnetic materials as there are still no
However because B=m0(H+M)
So sources of H are
possible which must also be sinks of M.
susceptibility and permeability
The magnetic susceptibility cm
a given point is defined as
Hence in the absence of magnetic materials B=m0H and in the presence of magnetic materials B=m0(1+cm)H.
H remains constant B
must change by a factor (1+cm) but this is also the definition of mr
and so we have
(Definition of mr)
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.
We take a cylinder as a Gaussian surface for the B-field.
As the height of the cylinder can be made infinitesimally small only the flux
through the ends of the cylinder need be considered.
Where B1n and
B2n are the normal components of the B-fields
Hence across any surface the normal component of B
For H we consider the
loop of length dL and of infinitesimal
The tangential component of H
is discontinuous by Jc (the
conduction surface current density) across any interface.
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 mr. We have
Where A is the area of
the solenoid, l is its length and n is
the number of turns per unit length.
Using these two equations to substitute for L and I in the equation for the magnetic energy
where the final term follows from B=m0mrH.
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
energy density =
If B and H
are not parallel then this result must be written in the form
These equations reduce to (1/2)m0B2
in the absence of magnetic materials.
In the presence of dielectrics we found that the electrical energy
density was given by
energy density =
Definition of relative permeability (mr)
LIH magnetic materials
Magnetisation (M) and
surface (Amperian) currents (Js)
– relationship between these
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 mr=1+cm
Boundary conditions for B
· Magnetic energy in the presence of magnetic materials
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