Current flowing through
a conductor produces a magnetic field. If the conductor is a long
straight wire, then the field is distributed over a large region
of space. 

If the wire is used to make a coil, the magnetic field is
concentrated into a smaller space and is therefore stronger. 



We could make some "educated guesses" at
the factors affecting the magnitude of the flux density inside a
solenoid: 

First, it must depend on the current flowing,
I. Let's guess that it is directly
proportional to the current (reasonable, as this is the case for a
long straight conductor). 

It also seems likely that it will depend on the number of turns
on the coil. 

However, consider the the first two coils shown below. 





The coil on the right has twice as many turns as that on the
left but is also twice as long. 



Now, how about these two coils? 





Again, the one on the right has twice as many turns as that on
the left but those turns are concentrated into the same space so one
might expect also a greater concentration of the magnetic field. 

So, we will suggest that the flux density
depends not just on the number of turns but on the number of turns per unit length. 

Again, let's suggest a direct proportionality. 

These two factors are relatively easy to test by experiment
and... wait for it... yes, our guesses are found to be correct! 

So we can write 



and so 



The constant turns out to be the permeability of the
medium inside the coil, m 



So the flux density inside a coil of length L having N turns and
carrying a current I is given by 



This equation gives the flux density anywhere inside an
infinitely long solenoid. 

In practice, as long as the solenoid is much longer
than its diameter, then the flux density is found to be of constant
magnitude over around 90% of its length (or more, depending on the
precise dimensions).. 

The equation above will give us this flux density. What about at
the end of the solenoid? 

To answer this question, first imagine yourself to be inside the
solenoid (ok, it's a pretty big solenoid now). 

You take out you binoculars and look to the left... what do you
see? 

Coils and coils and coils, off into the distance. 





Now look to the right... same thing. 

Remember that the current in all these
coils is helping to produce the field where you are standing. 

Now walk along to the end of the solenoid
and look to your left. 

Again, lots and lots of coils, off into the
distance. 





Now look to your right... nothing! 

Your conclusion? 

That's right, when we measure the flux
density at one end of a long coil, we find that it is just half
of the value at the centre. 



A graph of flux density against position
along the axis of a long coil (solenoid) looks something like this: 


