Electromagnetic Basics
1,2,3,4
In this section
we will look at the general electromagnetic principles which are widely employed
in engineering. This is a very short introduction to a complex subject. You should
find yourself a good book on magnetism and electromagnetism if you want to develop
a better understanding in this area. You can also find most of these concepts
examined in detail at Fizzics
Fizzle.
Electromagnetic
Fields and Forces
Before we look
specifically at the case of the coilgun it will be beneficial to briefly examine
the fundamentals of electromagnetic fields and forces. Whenever there is charge
in motion there is a corresponding magnetic field associated with it. This motion
can take the form of current in a wire, orbital electrons in a molecule or the
flow of a plasma etc. To help us with our understanding of electromagnetics we
employ the concepts of the electromagnetic field and magnetic poles. The differential
vector equations which describe this field were developed by James Clark Maxwell.
1. Systems of
Units 
Just to make life
difficult, there are three systems of units in popular use, namely the Sommerfield,
Kennely and Gaussian systems. Since each system has different units for many of
the quantities things can become confusing. I'll be using the Sommerfield System
as outlined below:
Quantity

Unit

Field 
H

Am^{1}

Flux 

weber (W)

Induction 
B

tesla (T)

Magnetisation 
M

Am^{1}

Intensity of Magnetisation 
I



Moment 
m

Am^{2}

Table
1. Sommerfield System of Units
2. BiotSavart
Law 
It is possible
to determine the magnetic field generated by a current element using the BiotSavart
Law.
Fig
2.1
where
H
is the field component at a distance r generated by the current i flowing in the
elemental length l
. u is a unit vector directed radially from l.
We
can determine the magnetic field generated by some basic current configurations
using this law. Consider an infinitely long wire carrying a current i. We can
use the BiotSavart Law to derive a general solution for the field at any
distance from the wire. I won't repeat the derivation here, any book on electromagnetism
will detail this. The general solution is:
Fig
2.2
The
field is circular and concentric with the current.
Another
configuration which has an analytical solution is the axial field of a current
loop. While we can develop an analytical solution for the axial field, it's not
possible to do this for the field in general. In order to find the field at some
arbitrary point we need to solve complex integral equations, this best done with
numerical techniques..
3.
Ampere's Law 
This
is an alternative method of determining the magnetic field due to a group of current
carrying conductors. The law can be stated as:
where
N is the number of conductors carrying a current i and l is a line vector.
The integration must form a closed line around the current. Looking at the infinite
wire again we can apply Ampere's Law as follows:
Fig
3.1
We
know that the field is circular and concentric with the current so H can
be integrated around the current at a distance r to give:
The
integration is very straightforward and shows how Ampere's Law can be applied
to provide quick solutions in some types of geometry. A knowledge of the field
pattern necessary before this Law can be applied.
4.
Field of a Solenoid
When
charge flows in a coil, it generates a magnetic field whose direction is given
by the right hand convention  Take your right hand and curl your fingers in the
direction of the current while extending your thumb, the direction of your thumb
now points to the magnetic north end of the coil. The convention for the direction
of flux has the flux emerging from a north pole and terminating on
a south pole. The field and flux lines form closed loops around the coil. Remember
that these lines don't actually exist as such, they simply connect points of equal
value. It's a bit like the contours on a map where the lines represent points
of equal height. The ground height varies continuously between these contours,
in the same way the field and flux from a coil are continuous (the continuum isn't
necessarily smooth  a disctete change in permeability will cause field values
to change sharply, a bit like a cliff face in the map analogy).
Fig
4.1
If
the solenoid is long and thin then the field inside the solenoid can be considered
almost uniform.
5.
Ferromagnetic Materials
Probably
the most well known ferromagnetic material is iron but there are other elements
such as cobalt and nickel, as well as numerous alloys like silicon steel. Each
material has a particular property which makes it suitable for its application.
So what do we mean by a ferromagnetic material? Put simply, a ferromagnetic material
is attracted by a magnet. While this is correct it's hardly a useful definition
and it doesn't tell us why the attraction occurs. The detailed theory of the magnetism
of materials is quite a complex subject involving quantum mechanics so we'll stick
to a simpler conceptual description. As you know, the flow of charge generates
a magnetic field so whenever we find charge in motion we should expect an associated
field. In a ferromagnetic material the orbiting electrons are arranged in such
a way as to generate a small magnetic field. Now this means that the material
is effectively composed of many tiny current loops which have their own magnetic
field. Normally the atoms are orientated in small groups called domains, these
are directed randomly throughout the material so there is no net magnetic field.
However if we apply an external field to the ferromagnetic material from a coil
or permanent magnet, the current loops try and align with this field  the domians
which are most aligned with the field 'grow' at the expense of the less well aligned
domians. When this occurs it results in a net magnetisation and attraction between
the material and the magnet/coil.
6.
Magnetic Induction and Permeability 
The
production of a magnetic field has an associated magnetic flux density, also known
as magnetic induction. The induction B is linked to the field by
the permeability of the medium through which the field penetrates.
where
0
is the permeability of free space and r
is the relative permeability. The unit of induction is the tesla.
7.
Magnetisation 
The
magnetisation of a material is a measure of it's magnetic 'strength'. Magnetisation
can be inherent in the material, such as in a permanent magnet or it can be generated
by external fields sources such as a solenoid. The magnetic induction in a material
can be expressed as the vector sum of the magnetisation, M, and
the magnetic field, H.
8.
Magnetomotive Force (mmf)
This
is analogous to electromotive force (emf) and is used in magnetic circuits to
determine the flux densities in the various circuit paths. Mmf is measured in
ampereturns or just amperes. The magnetic circuit equivalent of
resistance is reluctance which is defined as
where
l is the length of the circuit path,
is the permeability, and A is the crosssectional area.
Let's
look at a simple magnetic circuit:
Fig
8.1
The
torus has a mean radius of r and a crosssectional area A. The mmf is supplied
from a coil with N turns carrying a current i. The calculation of reluctance is
complicated by the nonlinearity of the materials permeability.
If
the reluctance can be determined then we can calculate the flux which exists in
the magnetic circuit.
9.
Demagnetising Fields 
If
a piece of ferromagnetic material such as a rod becomes magnetised, poles form
at the ends. These poles generate an internal field which tries to demagnetise
the material  it acts in the opposite direction to any field which is creating
the magnetisation. The result of this is that the net internal field can be much
smaller than the external field. The shape of the material has a large effect
on this demagnetising field, a long slender rod (high length/diameter ratio) has
a small demagnetising field compared to say, a stubby shape like a sphere. From
the perspective of coilgun design, this means that projectiles with a small length/diameter
ratio require a stronger external field to achieve a given state of magnetisation.
Take a look at the graph below, it shows the net internal field along the axis
of two projectiles  one 20mm long x 10mm diameter and the other 10mm long x 10mm
diameter. For the same external magnetic field there is a large difference in
the internal fields, the shorter projectile has a peak field about 40% that of
the longer projectile. This would very likely result in a significant difference
in the performance of the projectiles.
Fig
9.1
It
should be noted that poles only form when there is a discontinuity in the permeability
of the material. In a closed magnetic path like a torus, no poles form and so
there is no demagnetising field.
Sources:
Introduction to
Magnetism and Magnetic Materials, David Jiles
Electromagnetics
(Schaum's Outline Series), J.A. Edminister
Introduction to
Electromagnetism, M. Sibley
University Physics,
Harris Benson
Introductory Circuit
Analysis, Boylestad
