Repulsion between two magnets

Question

When we bring a bar magnet towards another bar magnet such that their similar poles are facing each other, the other bar magnet moves due to repulsion. How this is possible if magnetic force can't do any work?

Answer 1

This question deserves a clear and correct answer. The magnetic field by definition does not perform work. However, when you move a magnet work can be done by the electric field, $\vec \nabla \times \vec E = \partial \vec B / \partial t$. It is this electric field that does the work.

Because of the rather artificial distinction between electric and magnetic field this and similar questions keep appearing.

Answer 2

The sentence that magnetic force does no work is often mistakenly understood. A magnetic field exerts a magnetic force on a particle that is already in motion. However, the direction of this force is perpendicular to both the direction of the velocity of the particle and that of the magnetic field.

This force $F$ is quantified by $q\dot v\times B$, and by the principles of work done, there is no work done.

Now back to the problem, this seems to violate the principle of conservation of energy. However, notice that work is done to push the 2 magnets together, and this is in turn converted to potential energy, which is converted to work done via repulsion of the magnets.

Answer 3

This question is answered well in the top response to the link I put in the comments. I think the idea is, in the case of the bar magnet, that if there were no interactions between the particles making up the magnet, they would all move separately with accelerations perpendicular to their velocities, resulting in no work being done. It is the forces of constraint (i.e. Coulomb interactions) which hold the bar together that turn the larger force felt by the close end of the magnet into a net motion of the whole body. These are always allowed to do work.

Answer 4

A magnetic field does no work in the particular case of a freely-moving charged particle, because the force it exerts remains perpendicular to the direction of motion of the particle. However, this cannot be extrapolated to say that a magnetic field 'cannot ever do work'.

This is disproven by the case of a solenoid magnet, which can be turned on and off to lift a steel nail off of a surface (and drop it again). The solenoid field induces an opposing field in the nail, which causes an upward force to be exerted on the nail. When it moves upwards, clearly work is being done by the field.

The definition of work being done by a field is: if it is able to exert a force on an object, and that object then moves in the direction of the force, then work is being done. This is what is happening in the case of the solenoid magnet and your bar magnet example - work is being done by the magnetic field.

It is your claim that 'a magnetic field cannot do work' that is incorrect.

Edit:

One of the other answers claims that it is actually the electric field that is doing the work, not the magnetic field. I believe that is spurious, for the following reasons:

Consider the example of the two opposing bar magnets. I push them closer together and hold them there. Here, the magnetic field is not changing, so there is no electric field. Therefore, the potential energy is being stored entirely in the magnetic field. If I then release the magnets, in that t=0 instant, the magnets are not yet moving; therefore, again there is no electric field. Therefore, the motion of the magnets must be initiated by the magnetic field.

Clearly, the magnetic field is capable of storing potential energy, and then releasing that energy to initiate the motion of an object. If that doesn't count as 'doing work', then I'm not sure what does.

Even once the magnets are moving and there is an electric field present, the electric field cannot do work directly on the magnets, because they (presumably) have no net charge. Therefore, even in this case, the energy flow is as follows:

magnet1 -> magnetic field -> electric field -> magnetic field -> magnet2

The magnetic field is still involved in mediating the energy transfer between the two magnets, in which case I would say it is doing work.