Suppose a current carrying wire Is placed in vertical direction. A charged rod is placed is placed nearly horizontally.It is clear that magnetic field due to current carrying wire is inside the paper.So,We see here that magnetic force acts in upward direction on the charged rod. The force mg lies in downward direction.If we increase the magnitude of current,magnitude of field increases and force increases too. At some value of current, magnetic force exceeds the mg and it seems that the force is doing somekind of work and the rod lifts up. How is it possible when we know that work done by magnetic force is always zero. This question is troubling me for last couple of days. I am in a highschool student and we just have completed the magnetism chapter.
How is the magnetic field exerting a force on the charged rod? Magnetic fields exert forces on moving charges, not on stationary ones.
As @MikeV points out, if the magnetic field is changing then it induces an electric field. This can exert a force on the charged rod. But for a single current carrying wire to exert any significant induced force on a charged rod the rate of change of the current in the wire would have to be enormous.
I don't think in your question you are thinking about induced fields, though. You say
At some value of current, magnetic force exceeds the mg and it seems that the force is doing somekind of work and the rod lifts up
The "At some value of current" statement seems to indicate that you think the magnetic field exerts a force on the charged rod. It doesn't. You can't lift a rod this way.