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Actually I had posted a very similar question, but I wasn't quite satisfied with the answer so I am posting a new variety again.

Imagine an infinitely long wire carrying current $I_1$ from West to East. At a small distance $d$ above the wire there is another small current carrying wire of length $l$ carrying current $I_2$ from East to West (opposite to the direction of current in the below placed wire). Obviously they are magnetically repelling. And if the second wire (carrying current from East to West) is at rest the magnetic force must be equal to $mg$.

The magnetic force is upward and $mg$ is down. The magnetic force can be written as $$\frac{Uo I_1I_2}{2\pi d} l = mg$$ Now, if the magnetic force is greater than $mg$, the wire moves up. Now magnetic force is up and displacement is up too which means that work done by magnetic force should be positive. How is that possible when we know that work done by a magnetic force is always zero? Is it that an e.m.f. is induced which opposes the change?

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Mohammad Abid
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  • A constant - not varying in time - magnetic field doesn't do any work except the work done similar to a spring. So if one switch on the current in both wires a time varying magnetic field do work on both wires. After the electric currents get constant, the magnetic fields get constant too. – HolgerFiedler Aug 24 '15 at 09:56
  • Im sorry i dont get ur point – Mohammad Abid Aug 24 '15 at 10:15
  • Mohammad, the magnetic force on a charged particle is always perpendicular to the particle's velocity so the work done by the magnetic force on a charged particle must be zero. But note that, in the wire, the mobile electrons aren't free to, .e.g., execute circular motion in a uniform electric field, but are constrained by the surface of the wire. Consider the implication of that. – Alfred Centauri Aug 24 '15 at 10:44
  • cant the electrons exert a force on the walls the wire, which could lift the wire? – Mohammad Abid Aug 24 '15 at 11:01
  • Related: http://physics.stackexchange.com/q/108992/9887 – Alfred Centauri Aug 24 '15 at 14:24
  • The answer there is also not clear. ''The above explanation does not seem correct to me now, because one can extract work from the system of current carrying conductors very slowly, in which case the induced electric field will be negligible, while the force doing work is still great and given by the formula like ∫j1×B2d3x. Please remove the green sign of acceptance. The real answer to your question requires more insight'' this is what he mentions in the answer. – Mohammad Abid Aug 24 '15 at 15:19
  • @MohammadAbid, indeed that answer was not entirely right. The thing is the work on the wire is done by internal forces, electrons do work on the atomic lattice, external magnetic field just enables this, just as normal force enables us to rise in gravitational field, but the work is done by internal forces in human body. Check out my answer here: http://physics.stackexchange.com/questions/195032/positive-work-by-a-magnetic-field?lq=1 – Ján Lalinský Aug 26 '15 at 20:29

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Here is the misunderstanding:

Now, if the magnetic force is greater than mg, the wire moves up. Now magnetic force is up and displacement is up too which means that work done by magnetic force should be positive.

The statement that a magnetic field does no work is of a static magnetic field. You are positing a changing magnetic field by the word "greater" and "moves". To become greater energy has been supplied and it is not the static condition. That changing magnetic fields do work is evident :

An electromagnetic crane is a type of crane with an electromagnetic lift. Electromagnetic cranes are commonly utilized in lifting and moving various scrap metals. It does not have the mechanical 'pincers' of a regular crane, instead, it has a large flat magnet which draws the metallic materials to it.

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anna v
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Im somehow not able to comment on anna v answer so here goes my cmment

How isnt the magnetic field non static. Acc. To The relation the force is dependent on the the magnitudes of current. Lets suppose some constant current i1 is flowing in the below placed wire and we bring the wire 2 also carrying some dc constant current i2 and place it above the i1 wire. The moment we place the wire 2 in the field of 1 the repelling force is experienced by the wire 2. And for instance the magnetic force which is constant as i1 and i2 are constants exceeds the mg downwards the wire 2 moves up, the work is certainly done by static magnetic field. I dont see any other parameter which could change the magnitude of magnetic field and make it varying.

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