The average induced emf in conducting coil of the N loop is equal to the negative of N time the rate of change of magnetic flux through the coil.
ε = -N (∆∅)/∆t
The motional induced emf in a rod moving perpendicular to the magnetic field is given by
ε = -VBL → eq 1
Consider a conducting rod L moves
from position 1 to position 2 through the distance ∆x=x2-x1 in the time ∆t. The velocity of the rod is given by,
V = ∆x/∆t → eq 2
Putting this value in equ.1
ε= -VBL = – ∆x/∆t BL → eq 3
As the rod moves through the distance, the increase in the area of the loop is given by ∆A= ∆XL, T his increases the flux through the loop by ∆∅,
∆∅=∆AB=∆XLB → eq 4
Eq 3 becomes,
ε = – (∆∅)/∆t → eq 5
If three is a coil of N loop instead of a single loop, Then
ε= -N (∆∅)/∆t → eq 6
This shows induced emf will become N tims and vesting shows the direction of the induced emf is such that it opposes the change of flux.