# An Overview of EM Fields

Before
attempting to introduce the concepts of electric and magnetic fields,
it might be worth just clarifying some of the most basic concepts that
underpin electromagnetism, i.e. *charge, force, field, voltage, capacitance,
inductance, and flux*. Although stepping outside the timeline, it
is sensible to relate the definition of these terms in the context of
today’s units. As such, an elementary negative charge can be associated
with the electron and an elementary positive charge with the proton.
The units of charge are the Coulomb [C] that corresponds to 6.24 x 10^{18}
elementary charges, i.e. electrons or protons. Like charges exert a
repulsive force, while opposite charges exert an attractive force that
is defined by Coulomb’s law:

[1] F = K(q_{1}*q_{2})/R^{2}.

However, in order to explain this equation, we need to introduce the concept of an electric field, which will then allow us to expand our vocabulary to include energy, work, voltage and capacitance.

## Electric Fields

As indicated, *the idea of charge* is often initially considered to be a quantity
associated with a single particle, e.g. an electron, which can then
be thought to be surrounded by an electric field [E]. However, in practise,
the only way to measure this field is to detect the force exerted on
another charge in the form of a unit charge probe. The strength of this
electric field is proportional to the force at any point, which is defined
by Coulomb’s Law, as per [1] above. In the context of [1], the constant
[K] is determined by the *units used*, which we will define in terms of
present-day international (SI) units. As such, we can simply expand
the definition of [K] as follows:

[[2] K = 1/4πε_{0}

Note, for simplicity, this discussion will, by default, always assume
that all charge interactions take place in a vacuum. Therefore, [ε_{0}]
corresponds to the permittivity of a vacuum, which is also referred to
as the ‘*electric constant’ *. While this constant might not appear
to bring any additional clarity to [1], it does have some significance
in the development of *Maxwell’s equations*, especially in terms of the
propagation velocity [v] of an electromagnetic wave in vacuum. However,
first, we might want to just compare the similarity of [K] to the gravitational
constant [G] in Newton’s equation:

[3] F=G(m_{1}*m_{2})/R^{2}

It is clear from [1] that the force [F] has to be proportional to the charge [q], which leads to the definition of an electric field [E] being the force [F] per unit charge [q]:

[4] E = F/q = Kq/R^{2}

To avoid confusing electric field strength [E] with energy, the inclusion
of any energy parameters will carry a suffix denoting total [E_{T}],
potential [E_{P}] or kinetic [E_{K}] energy in the context
of this section. Given the similarity of [1] and [3], it is not so surprising
that the electric field [E] is also subject to the inverse square law,
as per gravity. However, it might be useful to consider the relative
strength of the electric field in comparison to a gravitational field
by taking the ratio of [1] and [3] and substituting for the masses of
a proton [m_{p}] and electron [m_{e}] and the unit charge
[e]:

[5]

Given this huge disparity in strength, it is clear that the atomic
structure is dominated by the electric field strength and not by gravitational
field strength, as initially assumed by the
*earliest atomic models*.
However, while gravity is miniscule in comparison to the electrostatic
force, most atomic structures are charge neutral, which explains why
on the macroscopic scale gravity is assumed to dominate planetary
and stellar motion; although some may debate this assumption - see
*Plasma Model*. Let us,
at this point, also introduce some of the other terms of reference by
considering the repulsive force acting between 2 equal signed charges.
To bring like-charges together will require energy or more accurately
work to be done, which we may define in terms of:

[6] Work [W] = Force [F] * distance [R]

We may now wish to consider the energy implications following on from [1] and [6], if we try to move an elementary charge [q] from infinity towards a unit charge [Q] of 1 coulomb, where both charges have the same sign, such that the force is repulsive:

[7]

Now we know all the units for the variables in [7] except for the
constant [ε_{0}], given that work [W] has the units
of energy. As such, we can we specify [ε_{0}] by
re-arranging [7] as follows:

[8]

We might also wish to use this example to compare the units of the variables that underline the nature of both voltage [V] and capacitance [C]. At a basic level, the definition of voltage [V] is measured in terms of the energy per charge and we have already defined a form of energy associated with moving charge [q]. However, in this context, energy is required to hold the charge [q] in position [R] against the repulsive force [F]. As such, the work energy expended has been converted to potential energy and therefore we might better describe voltage [V] as the potential energy per charge:

[9]

Let us also try to use this approach to provide some initial insight to the property called capacitance. Now different texts may describe capacitance in one of several ways, e.g. it is the ability of a body to hold an electrical charge or that it is a measure of the amount of electrical energy stored for a given electric potential. For the purposes of this initial definition, we shall simply defined capacitance as the charge per voltage, which can then be correlated to the units of capacitance, i.e. the Farad:

[10]

## Magnetic Fields

Let us now turn our attention to providing some basic introduction
to magnetic fields. The first thing to understand is that a magnetic
field is only present when a charge-particle is moving with velocity
[v_{1}] and only acts on another charge particle that is moving
with velocity [v_{2}]. The use of two velocity values is simply
highlighting that one moving charge particle is required to generate
a magnetic field, while another charge particle is being affected by
the magnetic field.

*There is an interesting aspect
to magnetic field in connection with special relativity. If [v _{1}]
and [v_{2}] are relative, then in some arbitrary frame of
reference, one of these velocities could be considered as zero. In
which case, the magnetic force would not exist. Maxwell’s equations
are invariant under Lorenz transformation and suggest a close
relationship with special relativity. However, further discussion of
this aspect will be deferred to a following section entitled 'Relativistic
Electrodynamics'.
*

The force on a charged particle due to a magnetic field is given by:

[11] F = qvB*sinθ = q**(v**x**B)**

The second form on the right is reflective of the cross product of
two vectors, which produces a perpendicular vector, e.g. force [**F**],
where [q] is the charge of the particle, [**v**] is its velocity
vector, [**B**] is the magnetic field vector and [θ] is the angle
between the direction of velocity and the magnetic field. The diagram
shows that the direction of the force is always perpendicular to both
the direction of velocity and the magnetic field and results in the
charge particle trying to move in a circular path of radius [r] around
the direction of the magnetic field due to the force [F]. We can determine
this radius [r] via the following equation:

[12]

On the basis that [ω=v/r], we can also express the angular velocity [ω] as follows:

[13]

In practice, the early pioneers experimented with the phenomenon
of electromagnetism by passing an electric current [I] through a wire.
Of course, we can *re-interpret this current* as the movement of ‘N’ charged
particles per second.

Note: The accuracy for the statement above might have to be interpreted in terms of the earlier finding of a discussion entitled 'Electricity' that suggested that electrons move 'slower than snails'.

In the context of a conceptually infinitely long straight wire carrying a current [I]. The Biot-Savart formula in [14] defines the magnetic field [B] at any point along this path:

[14]

As such, [14] has been used to shown the SI units of the magnetic
field [B]. However, this equation also allows us to define another constant
of electromagnetism, i.e. [μ_{0}], which is described as
the magnetic permeability of a vacuum. Again, knowing the units of all
other variables within [14] allows the units of [μ_{0}] to
be defined:

[15]

From a more intuitive perspective, most of us will have experienced
the effects of magnetism when playing with permanent magnets. One of
the things we quickly experience with permanent magnets is the *‘north-south’*
polarity; so now might be a good point to also introduce the issue of
magnetic dipoles and monopoles. A dipole is the name given to the normal
‘*north-south’* pairing associated with a permanent magnet.
However, while theoretical physics has predicted the possible existence
of magnetic monopoles, to date no evidence has been found to support
this conjecture. At the atomic level, a magnetic dipole can initially
be thought to result from the orbital motion of a charged electron around
a relatively static nucleus, as suggested by the diagram on the left
below. As such, this motion would essentially create a current loop,
surrounded by a magnetic field. However, quantum mechanics suggests
that it is the stronger effect of quantum spin, which actually leads
to permanent magnets, even though quantum theory then goes on to state
that electrons neither physically spin, nor orbit the nucleus.

While the models above come with several caveats, they attempt to
use the basic atomic model on the left as a building block, first as
an equivalent single dipole paring, which is then combined into larger
configurations that can ultimately be scaled to the macroscopic level
of a permanent magnet. In part, if you have ever broken a magnet into
2 pieces, you will realise that you end up with 2 smaller magnets which
can be *‘stuck’* back together in another composite north-south
configuration, as illustrated. Based on the model above, an atom might
be thought to form a magnetic dipole, but which in most material is
orientated at random. However, in a permanent magnetic material they
align in the same direction. So, although this introduction to magnetic
fields may be very limited, and possibly technically inaccurate on a
number of points, it is really only trying to illustrate the interaction
and interdependency of a magnetic fields on the electric field.