Electromagnetic Basics
1,2,3,4
10. Force on
Charged Particle 
So
how do we calculate forces on current carrying conductors? Lets start by looking
at the force on a charge moving in a magnetic field. I'll adopt the general approach
in 3 dimensions.
This
force is determined by the vector cross
product between the charges velocity, v, the magnetic induction, B,
and is proportional to the value of the charge. Consider a charge, q = 1.6x10^{19}
C, moving at 500m/s in a magnetic field of induction 0.1T, as shown below.
Fig
10.1. Charge in motion experiences a force
The
force experienced can be easily calculated as follows:
The
velocity vector is 500 i m/s and the induction is 0.1 k T so:
Obviously
if there is nothing resisting this force then the particle will be deflected (it
would describe a circle in the xy plane for the case above). There are plenty
of interesting things which can be achieved with free charged particles and magnetic
fields  you're reading this on one of them.
11.
Force on Current Carrying Conductor
Let's
now relate this to the force experienced on a current carrying conductor. There
are a couple of different ways of deriving the relationship.
We
can describe the conventional current as the rate of flow of charge 
now
we can differentiate the force equation above to give 
combining
these equations results in 
the
dl vector points in the direction of the conventional current. This
expression can be used to analyse physical arrangements such as the DC motor.
If the conductor is straight then this can be simplified to 
The
direction of the force is always at right angles to the flux and the current direction.
When using the simplified equation, the direction of the force is given by the
right hand rule.
12.
Induced Voltage, Faraday's Law and Lenz's Law
One
last thing we need to consider is induced voltage. This is just an extension of
the analysis of the force on a charged particle. If we take a conductor (something
with mobile charge) and move it at some speed, V, relative to a magnetic field,
the free charges will experience a force which will push them to one end of the
conductor. In a metal bar there will be a charge seperation where some
electrons have been forced to one end of the bar. The diagram below shows the
basic idea.
Fig
12.1 Voltage induced across a moving conductive bar
The
result of any relative motion between a conductor and a magnetic induction will
be an induced voltage generated by charge motion. However, if the conductor moves
parallel to the flux (the Z direction in fig X) then no voltage will be induced.
We
can look at another situation where an open planar surface is penetrated by a
magnetic flux. If we set up a closed contour, C, then any change in the flux linking
C will induce a voltage around C.
Fig
12.2 Flux bounded by contour
Now
if we introduce a conductive loop in place of C then the changing flux will induce
a voltage in this conductor driving a current around the loop. The direction of
the current can be found by applying Lenz's Law which basically states
that the effect acts in opposition to the cause. In this case the induced voltage
will drive a current which opposes the change in flux  if the flux decreases
then the current will try and maintain the flux (anticlockwise current), if the
flux is increasing then the current will try and oppose the increase (clockwise
current). Faraday's Law states the relationship between induced voltage, changing
flux and time 
The
negative symbol is a consequence of including Lenz's Law.
13.
Inductance
Inductance
can be described as the ratio of flux linkage to the current producing the flux.
For example, consider a wire loop of crosssectional area, A, carrying a current
I.
Fig
13.1
The
self inductance can be defined as
If
the loop is composed of more than one turn then the expression becomes
where
N is the number of turns.
It's
important to realise that the inductance is only a constant if the loop has an
'air core', ie is surrounded by air. When a ferromagnetic material is part of
the magnetic 'circuit' it introduces a nonlinear behaviour into the system which
results in a variable self inductance.
