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Obtain an expression for motional emf from Lorentz force

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Motional emf from Lorentz force:

Consider a straight conducting rod AB of length l in a uniform magnetic field \(\vec B\) which is directed perpendicularly into the plane of the paper. The length of the rod is normal to the magnetic field. Let the rod move with a constant velocity \(\vec v\) towards right side. When the rod moves, the free electrons present in it also move with same velocity  \(\vec v\) in \(\vec B\) . As a result, the Lorentz force acts on free electrons in the direction from B to A and is given by the relation

The action of this Lorentz force is to accumulate the free electrons at the end A. This accumulation of free electrons produces a potential difference across the rod which in turn establishes an electric field E directed along BA. Due to the electric field E, the coulomb force starts acting on the free electrons along AB and is given by

\(\vec F_E\)= -e \(\vec E\)  ........... (2)

The magnitude of the electric field \(\vec E\) keeps on increasing as long as accumulation of electrons at the end A continues. The force \(\vec F_E\) also increases until equilibrium is reached. At equilibrium, the magnetic Lorentz force \(\vec F_B\) and the coulomb force   \(\vec F_E\) balance each other and no further accumulation of free electrons at the end A takes place, i.e.,

vB sin 90° = E 

vB = E ……. (3) 

The potential difference between two ends of the rod is

Figure: Motional emf from Lorentz force 

V = El 

V = vBl 

Thus the Lorentz force on the free electrons is responsible to maintain this. potential difference and hence produces an emf

ε = Blv ….. (4)

As this emf is produced due to the movement of the rod, it is often called as motional emf

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