We label the circuit with the electric current in each branch as shown in the figure above. In addition, we set the origin of the x-axis to be at a distance L away from the left edge of the rail. Furthermore, we suppose that the bar is moving with a velocity of v at the position x away from the origin.

Since the resistance of rail is proportional to its length, rab = rfe ∝ (L+x), and rbc =red ∝ (L-x).

But rab+rbc=r, so it follows that rab=rfe=(L+x)2Lr,  and rbc=red    =(L−x)2Lr⁠.

There is induced motional emf in the bar...

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