A conductor moving in a magnetic field is able to induce an electromotive force.
Slidewire generator: Let’s consider a metal rod with length $L$ placed across a U-shaped conductor closing a circuit, moving at a constant velocity $\vec v$. The circuit is inside a uniform magnetic field (directed into the circuit’s plane). This induces an emf and a current along the circuit.
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Free moving charges in the moving rod experience a magnetic force $\vec{{F}}=q \vec{v}{rod} \times \vec{{B}}$. This force causes the positive (negative) free charges to move upwards (downwards), like an electric field of value $q E=q v{rod} B$. The effective voltage between the endpoints of the rod must be
$$ V_{a b}=\mathcal E=E L=v_{rod} B L $$#General Expression
It is straightforward to generalize the previous result for an arbitrary circuit.For an element $d\vec l$, the contribution $d\mathcal E$ to the emf is the component of the magnetic force parallel to $d\vec l$. Adding up these contributions we have
$$ \mathcal{E}=\oint(\vec v \times \vec B) \cdot d \vec{l} $$
Magnetic Materials