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According to my textbook and several other sources on the internet if we we move a rod in the direction perpendicular to a uniform magnetic field, the electrons will also move as per the direction given by the Lorentz force. After this it is written that due to accumulation of negative charges on one side and positive charges on the other side an electric field is developed and this electric field after a while becomes equal to magnetic force and that charges don't move any further and an EMF is developed.

Now my book has shown the rod moving on two metallic rails connected to a galvanometer. What I don't understand is that how would the aforementioned emf /potential difference be developed if the two end of rod are connected in a circuit, because wouldn't the charge that was to be accumulated just move past onto the metallic rails?

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As soon as any of that accumulated charge starts to move around the external circuit the electric field inside the wire, which produced a force on the charges equal in magnitude but opposite in direction to the Lorentz force, is reduced.
This means that the Lorentz force can now move some more charges to replace the accumulated ones that have gone for a walk around the external circuit.
Then the sequence repeats itself.
Thus you get a continuous flow of charge around the circuit.

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The emf produced due to the relative motion of the loop and magnet is given by E=VBl.where l is the length of the conducter and moving at speed V relative to the magnet.the emf can be calculated two different points in terms of magnetic force on moving electrons in a magnetic field and second term is rate of change of magnetic flux.both yield same result.

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