# What should be the direction of motion of a conductor in a magnetic field to produce motional emf?

Here's the textbook question: Twelve wires of equal length 'l' are connected to form a skeleton cube which moves with constant velocity v perpendicular to the magnetic field B. What will be the induced emf in each arm of the cube?

As per my understanding, motional emf is produced in a conductor that moves in a direction perpendicular to the magnetic field. Hence, each arm should develop a motional emf.

The book says - "Emf is produced only in the arms that are perpendicular to both $V$ & $B$. Thus, only EH, FG, AD, BC produce a motional emf, $\epsilon= BLV$ "

let us consider the branch EH velocity is directed +y and magnetic field is along +x so by definition F=qvXB positive charge accumulates at E thus giving it Some potential. point to note here net emf produced here will be zero as magnetic field is constant throughout so dφ/dt equals zero. other wires Example DC also produces potential but in a direction that is in the direction of width and not length so ΔL=0 aprox so transverse emf (SEE halls effect) will be almost zero even though it is not exactly zero. Same thing will happen to the wire DE say. Transverse EMF is produced so ΔL=0 aprox.

An EMF requires the force per unit charge of the source (in our case $\vec{v}\times\vec{B}$) and then you take the scalar product of that with the vector in the direction of the circuit element (in our case $d\vec{s}$).
If either $\vec{v}$ or $\vec{B}$ (or both!) point in the same direction as $d\vec{s}$ then the cross product $\vec{v}\times\vec{B}$ will be orthogonal to $d\vec{s}$, and so no EMF.