A conducting rod `MN` moves with a speed `v` parallel to a long straight wire which carries a constant current `i`, as shown in fig. The length of the rod is normal to the wire. Find the emf induced in the total length of the rod. State which end will be at a lower potential.

Explanation

The magnetic field induction due to current `i` is different at different sections of the rod, because they are at different distances from the wire.
Let us, first of all, subdivide the entire length of the conductor `MN` into elementry sections. Consider a section (shown shaded in the figure (b)) of thickness `dx` at a distance `x` from the wire. As all the three, `v,B` and `(dX)` are mutually normally to each other, so the emf induced in it is `de=Bvdx`. ( from `N` to `M` by Fleming's right hand rule )

For the sections, the induced emf is in the same sence, (i.e., from `N` to `M`)
`:.` Total emf induced in the conductor is
`e=intde=int_(b)^(b+a)Bvdx`
Substituting for `B=(mu_(0)i)/(2pix)`, the above equations gets changes to `e=int_(b)^(b+a)(mu_(0)iv dx)/(2pix)e=(mu_(0)iv)/(2pi)[lnx]_(b)^(b+a)` or, `e=(mu_(0)iv)/(2pi)ln ( 1+a//b)`