Can stationary permanent magnet do work?

[goran d]

We have a permanent saturated magnet. And a coil wound around it. The current produces magnetic field in same direction as the magnet. Now we know that the energy of magnetic field is proportional to the square of the magnetic induction.
E₁=kB₁²
E₂=kB₂²
E_total=kB₁²+kB₂²+2kB₁B₂
We have an extra energy term when we add the fields. It grows proportionally with the current in the coil. The extra energy has to come from somwhere, or does it?
The magnet appears to be doing work in increasing the field.
This is fully compatible with Poynting Theorem. The integral of the dot product of the electric field and current density is equal to the increased energy. However, what that force is trying to do is to spin down the magnetic domains, which it can't do. So the electric force simply has no effect on the domains. Where does the extra energy come from?
If we increase the current to a very large value, can we then cool down the magnet, it to lose its magnetism, and it gives a much larger energy "boost" to the coil than the energy we spend?

 

[Hesch]

Now we know that the energy of magnetic field is proportional to the square of the magnetic induction.

Just a comment:

E_dens = ½ * B * H [ J/m³ ] = ½ * B² * μ

You say that the permanent magnet is saturated, so when extra H-field is added by the coil, μ → μ₀ ( almost immediately ).

 

[Jonathan Scott]

Consider what would happen if the coil magnet started off separate from the permanent magnet. You would have to do work to bring it into position, because being the same way round it would repel the permanent magnet. So the total energy would now be more than the sum of the energy of each separate field.

On that basis, one would expect to need more electrical energy to create the same amount of current (and hence that magnetic field) in the coil when it is wound round the permanent magnet than it would when it is separate.