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Flux Density inside a Long Coil (Solenoid)
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:   
   
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