Lectures 15 and 16 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  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.

 

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 mr of the material is defined as

mr=Lm/L0

Because the magnetic flux F=LI this flux changes as does the average magnetic field defined as <B>

mr=Fm/F0=<B>m/<B>0

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.

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 dt 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/dt             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.

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.

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.

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 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.

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

B=B0+Bm

Ampres 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

 

We 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 along  M. 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)

 

dlJs=dLJscosa=dLMcosa=MdL

using the fact that M=Js

Hence the circuital law becomes

 

or combining the two line integrals 

                            (A)

this result gives the modified form of the circuital law in the presence of magnetic materials.

The H-field

Because the quantity B/m0-M occurs quite often it is given a special name magnetic field strength or H-field, symbol H.

 

      units Am-1

       (B)

  The circuital law for H is (from (A))

 

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/m0. In any situation H is given by the corresponding formula for B divided by m0. For example for an infinitely long wire

 

Because H can only arise from conduction currents these equations are also valid in the presence of magnetic materials.

The equation B=0 is unmodified in the presence of magnetic materials as there are still no magnetic monopoles.

However because B=m0(H+M)

B=0=m0(H+M) H=-M

So sources of H are possible which must also be sinks of M.

Magnetic susceptibility and permeability

The magnetic susceptibility cm at a given point is defined as

M=cmH

But B=m0(H+M)= m0(1+cm)H

Hence in the absence of magnetic materials B=m0H and in the presence of magnetic materials B=m0(1+cm)H. As H remains constant B must change by a factor (1+cm) but this is also the definition of mr and so we have

mr=1+cm               (Definition of mr)

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.

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.

 


For B   and applying to the ends of the Gaussian cylinder

B1n-B2n=DBn=0

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

(H1t-H2t)dL=DHtdL=Ic DHt=Ic/dL=Jc

The tangential component of H is discontinuous by Jc (the conduction surface current density) across any interface.

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 mr. We have

L=Am0mrn2l and B=m0mrnI

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 general one

Magnetic 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

 

Electrical energy density =

 

Conclusions

        Definition of relative permeability (mr)

        LIH magnetic materials

        Magnetisation (M) and surface (Amperian) currents (Js) relationship between these

        B-fields in magnetic materials

        Ampres 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 and H

        Magnetic energy in the presence of magnetic materials

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