__Maxwell's Equations__

There are two
different types of field, electric and magnetic. The relationship between these
two fields and charges and currents can be summarized in just four statements
or equations. These are known as Maxwell's equations. An extremely clear and
very detailed description of Maxwell's equations, how you derive them and how
you apply them using both basic mathematics and vector calculus is given by
Feynman in the early chapters of volume 2 of his famous books The Feynman
Lectures on Physics (Ref 3a). In words, Maxwell's
equations are :-

1a) For any closed surface the average normal component of the

electric field through the surface times the surface
area

EQUALS

The sum of the charges enclosed divided by ε_{o}

Where ε_{o} is just a constant to ensure the units
come out correctly. The sign of the charges is also important so if equal
number of positive and negative charges are enclosed the sum of these charges
is zero. The normal component is the field component at right angles to the
surface.

If we consider
any surface bounded by a loop (similar to considering a drawstring bag as the
surface and the perimeter of the bag opening, however wide, as the loop) :-

2a)
Around any loop in space the average tangential
component

of electric field times the distance around the loop

EQUALS

The rate of change of the average normal component of

magnetic field through the surface of the loop times the

surface area.

3a)
For any closed surface the average normal component of
the

magnetic field through the surface times the
surface area

EQUALS

Zero

This is similar
to equation (1a) but for the magnetic field rather than the electric.

4a)
Around any loop in space
the average tangential component

of the magnetic field times the loop length times c^{2}

EQUALS

The rate of change of the average normal component

of electric field through the loop times the loop area

MULTIPLIED BY

The total electric current through the surface divided by εo

Where c^{2}
is the velocity of light squared. This is similar to equation (2a) for the
electric field but involves an extra current term.

In using the
above definitions a closed surface is the bounding surface of any volume of
space ie the outside surface of a sphere or the outside of a cube or in fact
the outside of any object you care to imagine. The normal component of a field
is just the component at right angles to the surface being considered.

The above
equations are all that is required to explain the relationships between the
electric and magnetic fields and it is amazing how much complexity can arise
from such a seemingly simple set of relationships. Also it is surprising that
it is so difficult to find the various field configurations that can be formed
which still satisfy all the above. To this end it is much easier to put the
expressions into mathematical equations which have a more precise meaning and
allow the various terms to be manipulated further. There are a number of
different mathematical ways of doing this but the one chosen for this web site
is vector calculus. Using this notation some of the phrases used above take on
a symbolic form. For example:-

The average
normal component of a field (electric E or magnetic B) per unit volume is
denoted as the divergence and abbreviated to divE or divB, so the total
electric field out of a volume (dV) is divE.dV .

The charge per
unit volume is denoted by ρ, so the total charge in a volume (dV) is just ρ.dV . Using these symbols allows equation (1a) to be
rewritten as:-

divE.dV = _{}

or divE = _{} ________________(1b)

The average
tangential component of an electric field around any loop is the normal
component of curlE times the area of the loop. The usual convention is that the
magnitude of this curl is shown by a vector normal to the surface area of the
loop. However, it is possible to go around the loop either clockwise or
anticlockwise. The convention always adopted is that if you curl your right
hand around the loop with your fingers pointing in the direction you have
decided is positive, then your thumb points in the direction of the positive
normal to the surface. i.e. If you have a circulation
in the direction of your fingers, the curl is a vector in the direction of your
thumb.

The rate of
change of a quantity in calculus is shown as _{} ,
hence the rate of change of the average normal component of the magnetic field
through the surface of the loop of area (dA) is written _{} .
With the convention adopted for the curl direction this rate of change is
negative. Using these two expressions we can

write the second of Maxwell's equations as:-

curlE dA = - _{}

or curlE = - _{} ________________(2b)

Using similar
notation as for equation (1b), equation (3b) becomes:-

divB = 0 _________________(3b)

Equation (4a)
becomes:-

c^{2}. curlB dA
= _{} + _{}

and as I = j dA, where j is
the average normal component of current per unit area through the surface. Then:-

c^{2}. curlB = _{}
+ _{} __________________(4b)

The last
equation involves a flow of current. Often we know the currents and charges and
can use the equations to calculate the fields that they produce. In problems
involving the propagation of fields in a vacuum the current and charges are
taken to be zero and equation (4b) simplifies to:-

c^{2}. curlB = _{} ___________________(5b)

The above
equations numbers (1b), (2b), (3b) and (4b) are Maxwell's equations in
mathematical form. In a charge free vacuum we can just use equations (1b),
(2b), (3b) and (5b) and these enable us to calculate the propagation of any
electromagnetic field.