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EMF Induced in a Conductor Moving through a Magnetic Field
When some other type of energy is converted into electrical energy, we use the term emf to refer to the quantity of energy given to each Coulomb of charge.  
   
For example, when a battery converts chemical energy into electrical energy the quantity of energy given to each Coulomb of charge by the chemical reactions is called the emf of the battery.  
Similarly, when mechanical work is done turning an electric generator, the quantity of energy given to each Coulomb of charge is the emf of the generator.  
   
The term “emf” originally came from the phrase “electro-motive force”.  
This is now considered (by some pedants!) to be an inappropriate term as emf is a quantity of energy not a force.  
However, the abbreviation is still used.  
Having said all that, it should be clear that emf is another way of saying voltage...  
   
If a conductor moves so as to cut through lines of magnetic flux, a potential difference (voltage) is generated in the conductor. This p.d. is referred to as an induced emf.  
   
Consider the situation shown in the diagram below.  
 "G" represents a galvanometer.  
   
 
   
A long loop of wire is partly in a uniform magnetic field, of flux density B.  
We will consider a length, L of this part of the wire.  
The wire is moved, at constant speed, through a short distance, Δs, in time Δt, perpendicular to the flux lines.  
   
As described here, any electrons in the wire will experience a force at right angles to the field and their direction of motion through the field.  
Any free electrons will move round the circuit, as a result of this force and the galvanometer will show a deflection (but only while the wire is moving).  
Suppose that a total charge ΔQ (possessed by a bunch of those free electrons) moves past any point in the circuit in time Δt.  
Now, the energy given to those electrons to push them round the circuit comes from the work done the force F (see diagram).  
That is, it comes from whoever (or whatever) is pushing the wire through the field.  
The work done by this force is  
 
and this work is therefore equal to the energy given to ΔQ Coulombs of charge.  
The work done per unit charge is  
 
and, as stated in the introduction above, this is equal to the emf induced in the wire.  
 
Notice that we now have a situation in which a wire carrying current is surrounded by a magnetic field so as described here, this wire will experience a force.  
This force must act in the opposite sense to the force shown in the diagram.   
If you have any doubts about this, think what would happen if it were not true... and keep in mind the law of conservation of energy...  
If the wire moves at constant speed, this force must also be equal in magnitude to the force F in the diagram.  
   
Therefore we can write  
 
Combining the two previous equations and remembering that current is equal to rate of flow of charge (ΔQ/Δt), and that the speed of the wire through the field is given by Δs/Δt, we can write  
 
which reduces, rather nicely, to   
 
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