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The electrons in a conductor are free to move. So when a conductor moves in a magnetic field, the electrons experience a force and the e.m.f. is just the work done by the magnetic force. We illustrate this by means of an example.
Conductor moving in magnetic field
Consider a metallic rod sliding in rails with velocity \(v\) as shown in figure below. The charge carriers in the conducting rod free to move inside the rod. They experience a force $qvB$ directed from $A$ to $B$ \begin{align*} \epsilon = & \int \vec{F}~\overrightarrow{dl} = \int_{AB} \vec{F}~\overrightarrow{dl}\\ = & vBl = \frac{d}{dt}(xBl)\\ = & \text{rate of change of flux} \end{align*} 
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4727:Diamond Point
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