As long as the rod moves at a constant speed, all the work done by external person the rod is dissipated as heat in the resistor. As per Work-energy theorem, there is no change in kinetic energy of rod.
Here, force applied by external agent isF=F=BIl.
However, if applied external force is doubled then a part of external power increases the velocity of rod the rest is dissipated as heat.
Obviously, Lenz's law is not satisfied under such circumstances. Further,P=E2/R=B2l2/RP=E2/R=B2l2/R. Thus, if resistance R is doubled then power required to maintain the constant velocityv0v0becomes half.