**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 *L _{0}*.
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

*m*_{r}*=L _{m}/L_{0
}*

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

*m** _{r}*=

Unlike *e** _{r}* for dielectrics which always has a value >1 (

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*t* of a magnetised material will possess a
magnetic dipole moment d**m**. The
magnetisation is defined as the magnetic dipole moment per unit volume

**M**=d**m**/d*t
*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 J_{s}

Consider
a small element of the magnetised material in the form of a cylinder of
cross-section d*S* and length d*l*. The magnetic dipole moment of this element *M*d*S*d*l* follows from the above definition of the magnetisation.

The equivalent surface
current density is *J _{s}* so
that the total surface current is

·
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 **B*** _{0}*
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 **B _{m}** is produced which results from the existence of the
Amperian surface current

The total **B**-field is now
the sum of the original plus the new field

**B**=**B _{0}**+

Ampère’s
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 *I _{C}*
and Amperian

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 *I _{B}*
and the magnetic material

In order to
evaluate the circuital law in this case we need to calculate the contribution to
*I _{M}* as the path passes
through material

d*lJ _{s}*=d

using the fact that *M*=*J _{s}*

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**/*m _{0}*-

units Am^{-1}

**H** is (from (A))

from which the differential form may be derived

**Ñ****´H**=*J _{c}*

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**/m_{0}.
In any situation **H** is given by the
corresponding formula for **B** divided
by m_{0}.
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 Ñ×**B**=0
is unmodified in the presence of magnetic materials as there are still no
magnetic monopoles.

However because **B**=*m _{0}*(

Þ
Ñ×**B**=0=*m** _{0}*Ñ×(

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

Magnetic
susceptibility and permeability

The magnetic susceptibility *c _{m}*

**M**=*c _{m}*

But** B=****m _{0}**

Hence in the absence of magnetic materials **B**=m_{0}**H** and in the presence of magnetic materials B=**m _{0}**

*m** _{r}*=

Boundary
conditions for B and H

At a boundary between two different magnetic materials there may be
both a surface conduction current *J _{c}*
and an Amperian surface current

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 _{1n}*-

Where *B _{1n}* and
B

Hence across any surface the normal component of **B**
is continuous.

For **H** we consider the
loop of length d*L* and of infinitesimal
height

From

(*H _{1t}*-

The tangential component of **H**
is discontinuous by *J _{c}* (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)*L**I** ^{2}*.

Consider a solenoid which is filled with a magnetic material of
relative permeability *m** _{r}*. We have

*L=Am _{0}*

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**=*m*_{0}*m*_{r}**H**.

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)*m _{0}*

In the presence of dielectrics we found that the electrical energy
density was given by

*Electrical
energy density* =

Conclusions

·
Definition of relative permeability (*m _{r}*)

·
LIH magnetic materials

·
Magnetisation (**M**) and
surface (Amperian) currents (*J _{s}*)
– 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 *m _{r}*=

·
Boundary conditions for **B**
and **H**

· Magnetic energy in the presence of magnetic materials