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Registered Member #190
Joined: Fri Feb 17 2006, 12:00AM
Location:
Posts: 1567
This is what I read:
"Earnshaw's Theorem The proof of Earnshaw's theorem is very simple if you understand some basic vector calculus. The static force as a function of position F(x) acting on any body in vacuum due to gravitation, electrostatic and magnetostatic fields will always be divergenceless. divF = 0. At a point of equilibrium the force is zero. If the equilibrium is stable the force must point in towards the point of equilibrium on some small sphere around the point. However, by Gauss' theorem, / / | F(x).dS = | divF dV /S /V
the integral of the radial component of the force over the surface must be equal to the integral of the divergence of the force over the volume inside which is zero. QED!"
Can anyone explain this in more simple terms? I understand the concept. I just don't know vector calculus too well to appreciate why these two terms are set equal and why this is an impossibility. Thanks
Registered Member #29
Joined: Fri Feb 03 2006, 09:00AM
Location: Hasselt, Belgium
Posts: 500
Try to imagine "balancing" two repelling magnets so that one stands above the other... It slips all around and tries to flip over and you just cannot do it! This is because in order for the magnet to levitate, you must find the point where all torques vanish and the repelling force exactly cancels gravity. There is only one theoretical point that will satisfy this. Any infinitesimal deviation results in instability and the magnet flips over and probably squashes your fingers! :-@
But, if the top magnet is spinning like a gyroscope....it gets much more interesting!
Registered Member #190
Joined: Fri Feb 17 2006, 12:00AM
Location:
Posts: 1567
I am aware that by spinning it is possible to violate the theorm and balance the magnet, but that one deviates from this discussion.
Does the theorm state that there is no point in space where the torques are balanced? What do each of those integrals represent and what is the rational to setting them equal? In case it is not obvious, I never studied this topic in college physics.
Registered Member #29
Joined: Fri Feb 03 2006, 09:00AM
Location: Hasselt, Belgium
Posts: 500
There is one point and one point only. But it is an unstable equilibrium. Any infinitesimal deviation, and the magnet flips over. The system is in a state of maximum potential energy at this point and will try to seek the minimum potential energy point (where the two dipole moments line up, head-to-tail...with the minimum distance between them..
I have tried to give a non mathematical description... If you want mathematics, I can do that for you too...but you really have to want it!!!
Registered Member #29
Joined: Fri Feb 03 2006, 09:00AM
Location: Hasselt, Belgium
Posts: 500
Hi Smooth!
Scribbled a few notes.. If you can get your head around Gauss's theorem, it is not too hard. Basically for the system to be stable, any deviation from the stationary position (position where forces vanish), must cause a force that restores the particle to the stationary point. This is not possible in a static electric/magnetic field.
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Earnshaw's Theorem - anyone want to explain better
