Consider a region where the magnetic field is expressed as B = k B₀ L y , with L being a constant length. A conducting rod of length L is positioned along the Y-axis from the origin to the point (0, L, 0) . Determine the induced emf between the ends of the rod if it travels with a velocity v = v₀ i .

Step-by-step solution

The motional emf induced in a small element d l of the rod is given by d = ( v B ) d l . Here, v = v₀ i , B = B₀ L y k , and d l = dy j . Evaluating the cross product: v B = (v₀ i ) ( B₀ L y k ) = - B₀ v₀ L y j The induced emf in the small element is: d = ( - B₀ v₀ L y j ) (dy j ) = - B₀ v₀ L y dy The total induced emf is the magnitude of the integral of d from y = 0 to y = L : = | ₀^ L - B₀ v₀ L y dy | = B₀ v₀ L [ y^2 2 ]₀^ L = B₀ v₀ L ( L^2 2 ) = B₀ v₀ L 2