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If in a uniform magnetic field a conducting rod moves on a set of joined friction less rails which are perpendicular to the direction of magnetic field, a potential is developed and current flown this current generates heat, the mechanical energy required to move the rod is converted to heat energy in wires.

Now if a battery of EMF equivalent to the induced emf is joined with opposite polarity, no current will flow and hence no heat will be dissipated. How in this case the mechanical energy required to move this rod is conserved ?

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in 1st case : you do not have a battery, current flows and creates a backward force on the wire given by F = IlB, to overcome this force you need to apply a force and do work against it. This work done gets converted to joules heat.

However in case 2 since no current flows , no force is applicable and F = 0, since there is no force to work against there is no work done in 2nd case and hence no mechanical energy to conserve, if the rod is moving it continues to move without slowing.

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