# Non-classically, do magnetic fields do work?

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1. Jun 3, 2015

### Nate Wellington

I know this question has been beaten to death, but I haven't seen a response that clearly (to me) answers the following:

1. Magnetic fields *can do work* on intrinsic dipoles, right? (e.g. two electrons can do work on one another via their intrinsic spin).

2. Magnetic materials can do work on one anther, right? "Bound currents", in this case, are fictitious--it is the intrinsic moments of the electrons that provide the magnetization, and there is no reason to believe they cannot do work, right?

Thanks!

2. Jun 8, 2015

### Hesch

A magnetic field contains energy density = ½*B*H [ J/m3 ]. Thus the energy in some (small) airgap between two magnets will contain the energy:

E = (½*B*H) * V, ( V is the volume of the airgap ). This can be rewritten:
E = (½*B*H) * A*s, ( A is the cross section area of the airgap, s is the width of the airgap ).

The energy density is much higher in the airgap than in the magnets because the relative permeabilty, μr = 1 in air, but μr ≈ 1000 in steel. So the strength of the H-field is much higher in air than in steel.

The nature wants to get rid of magnetic energy, converting it to another type of energy. This can be done by letting the magnets close up (due to attraction), thereby substituting airgap by magnet (with lower energy density ). That's the answer to: Why are two magnets attracting each other.

During this "closing up" the magnets, there will be an attraction force:

F(s) = (μr,steel - μr,air )*dE(s)/ds = (μr,steel - μr,air ) * ½*B(s)*H(s)*A [N].
The work done by closing up the magnets = s F(s) ds = E*(μr,steel - μr,air ).

Last edited: Jun 8, 2015
3. Jun 12, 2015

### Nate Wellington

I think I am trying to answer a more "fundamental" question than that, one that doesn't require the use of permeabilities, etc.

David Griffiths states that magnetic fields can "do no work." It seems to me that: (1) if you have just two pure magnetic dipoles--electrons--the magnetic field obviously will do work work on them, and that (2) this isn't actually fundamentally different than what happens with two bar magnets, for example. Is that correct?

My understanding (as of right now) is that classically magnetic fields do no work, since you treat magnetic dipoles as current loops and attribute any work done to the electric field done in reducing the net size of those combined currents ("bound currents"). From a quasi-quantum mechanical level, where electrons are dipoles, this is not the case, and magnetic fields absolutely do work. Is this wrong?