If there are three electromagnets placed horizontally in a triangle pushing upwards, then there will be a dip in the middle and above the triangle where all three fields meet. Right? A metal sphere, completely in theory, would be able to rest there. Now, below these three electromagnets sits a single, much more powerful electromagnet that pulls the small metal sphere downwards, towards it. If you moved the entire four-magnet gig around, the metal sphere would be suspended, caught in the middle of the four magnetic fields. Now, rip this apart, please. I have been told by people I trust that this wouldn't work, that the fields would make the others go crazy, I'm just curious as to why.
Magnetic fields (and electric fields) work by superpotential. If magnet 1 gives field \(\mathbf{B}_{1}\) and magnet 2 gives field \(\mathbf{B}_{2}\) then having both magnets gives the field \(\mathbf{B}_{1} + \mathbf{B}_{2}\).
You do not give enough information about your "metal sphere" to reply uniquely, but here are the two main possibilities. If it is fero magnetic sphere, and exactly equally distant from the three identical magnetcs it could be in unstable equlibrium on the axis, but the slightest movement towards anyone of them would accelerate it to contact with that magnet. This assumes the unstable equlibrium point is slightly below the plane of the magnets as some net upwards magnetic force is required to balance the donward force of gravity. It is also unstable against gravity in that even if it always remains exactly on the symetry axis (equal distant from the three magnets) any slight down axis move will allow gravity to capture it (make it fall). Any up axis tiny motion will also cause it to fall down the axis, but with more complexity. I.e. the up axis motion will be first stopped by gravity as the up axis magnetic force is going to zero as it approaches the plane of the magnets, but force of gravity is essentially constant. Then it will start to fall back to (and thru) the unstable equlibrium point (because when it gets there, it will have speed downward). ----------- If the sphere is a super conductor and the magnets are strong enough, then there is a stable point for the sphere on the axis some what above the plane of the magnets. Note I have ignored your fourth on axis magnet as its effect resembles gravity, but of course is not of constant strength.
Opps. I've been typing that word so many bloody times over the last week I practically think in terms of cohomologies now....
You are forgiven! I was ill last week, and the low point was being unable to sleep because I was dreaming about a counterexample to a well known result in functional analysis. I blamed my fever, but my wife seems to think it might have been related to my bed time reading (which is currently an edition of "topological methods in nonlinear analysis").
ha ha! When I was an undergraduate I used to dream about integration quite a lot. It all stopped until recently when I dreamed about an expanding space where everything was being stretched. It was quite disturbing actually...
