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The Laws of Electro-Magnetic Induction
Lenz's Law
Consider a "thought experiment"...  
Imagine pushing a bar magnet towards a coil of wire as shown below  
   
   
We will consider the coil to be fixed in place.  
As the magnet is moving, the coil will be "cutting through" the lines of magnetic flux and so an emf will be induced in it.  
The ends of the coil are connected so current can flow as a result of this emf.  
In which sense will the induced current flow?
Suppose it flowed in such a sense as to produce something like the south pole of a magnet at the left hand end (see here for fields produced by coils).  
   
   
If this were the case then the coil would attract the magnet, so it would move faster, so the induced current would be stronger, so it would be attracted more strongly so... you get the idea...  
   
   
The magnet would pass through the coil.   
At half way through, the polarity of the magnetic field would change, the magnet would therefore still be getting faster and faster... finally disappearing, never to be seen again.   
Guess what... this does not happen.   
   
Conclusion  
When the magnet is pushed towards the coil, any induced current which flows must be in such a sense as to produce a force which repels the magnet.   
   
It is the existence of this force which is responsible for the energy conversion  
If you are pushing the magnet towards the coil, you are working against this repulsion and this results in electrical energy being generated.   
   
   
Btw, this method of proving something by showing that its opposite cannot be true is can be very useful and is called... well, something in Latin... google it!  
   
Many similar examples could be found to show the same basic idea which is now called Lenz's law (after the scientist Heinrich Lenz), stated as follows   
When electro-magnetic induction occurs, any induced current will flow in such a sense as to oppose the change producing it  
This is, in effect an electro-magnetic version of the law of conservation of energy.   
You can actually feel this effect directly if you can get hold of something like a bicycle dynamo.   
Try rotating the dynamo by hand when it is not connected (electrically) to anything.  
Try to turn it as fast as possible.    
Then connect a short circuit to the output of the dynamo (eg a piece of copper wire).  
Again, try to turn it as fast as possible.   
You will find that it is noticeably harder to turn with the short circuit in place.   
   
Faraday's Law (The Faraday-Neumann Law)  
As described here, when a conductor moves so as to cut through lines of magnetic flux, an emf is induced in it.   
It was shown that the induced emf depends on the speed of the conductor moving though the field (as well as the flux density and the length of the conductor).   
Michael Faraday conducted experiments in which he placed various conductors (usually in the form of coils of wire) in changing magnetic fields  
One of the quantities he varied was the rate at which the magnetic field strength varied  
He came to the following conclusion, which is now called Faraday's law of e.m.i. (for fairly obvious reasons):   
The induced emf is directly proportional to the rate of change of the magnetic flux linking the conductor.   
Both the laws of e.m.i. can be expressed in one mathematical statement:   
   
where the simple inclusion of a minus sign expresses Lenz's law.   
This mathematical statement is also called Neumann's law of e.m.i.   
At first sight, though this statement might seem perfectly reasonable, it doesn't look much like the equation found here, namely 
   
   
However, referring to the diagram above (showing a conductor of length L moving a distance Δs in time Δt, cutting through a uniform magnetic field):   
   
which rather nicely shows that the two apparently different situations   
1. a conductor cutting through lines of flux and   
2. a conductor surrounded by changing magnetic flux   
are, in some ways, equivalent.   
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