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Let's say we have a magnet stuck to a metal bar, suspended above the ground. If I attach a paperclip to the magnet, where is the energy to hold the paperclip coming from (against the force of gravity), and for how long will the paperclip remain there - will it remain there forever?

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When an object is hanging from a hook, where is the energy to hold the object against the force of gravity coming from? –  Sparr Nov 10 '10 at 0:28
    
@Sparr, good analogy. –  Thomas O Nov 10 '10 at 0:31
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2 Answers 2

up vote 12 down vote accepted

When the magnet attracts and moves the paperclip, moving it in the gravitational field of the earth, the energy comes from the potential energy that [in that case] we can associate to the magnetic field.

By attracting the paperclip you increase the gravitational potential energy of the paperclip but you reduce the one it has due to the magnetic field.

You do not need energy to hold the paperclip. As an analogy: when you hold something in your hands, you feel like you are "working" even in a static case; but in mechanics, work = exchange of energy is defined as $W = \int \vec{F} d\vec{s}$.

When the situation is stationary, no energy exchange is involved, no force does no work at all, so it can stay forever.

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Exactly. It is a very important point that no work is done unless an object actually moves a distance. (p.s. Might want to surround your equation with $ ... $ so the LaTeX shows up?) –  Noldorin Nov 9 '10 at 19:16
    
Latex is not working for this equation ? It looks OK for me. –  Cedric H. Nov 9 '10 at 19:18
    
I think your first sentence may be a little misleading, since there is no energy (work) involved. –  David Z Nov 9 '10 at 20:28
    
@David: I edited to reflect what I meant. I think like that it is OK. –  Cedric H. Nov 9 '10 at 20:35
    
Sure, I guess that's better. –  David Z Nov 9 '10 at 20:37
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The paperclip is not moving relative to earth - this means that no energy is being spent by magnet to hold it, so it can hold the paperclip as long as the magnet has magnetic properties.

As a contrast, when you hold something with your hands, energy is being spent by your muscles not to hold the object, but to remain contracted against gravity. This is because natural state of muscle cells is to be stretched or contracted as gravity tells them to, not against it.

By analogy, electromagnet spends energy not to hold things, but to to have magnetic properties, because its natural state is not magnetic. So natural magnet does not spend any energy, since its natural state is being magnetic.

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Actually, an electromagnet "spends energy" due to the resistance in its wire, not due to maintaining a magnetic field. A superconducting loop, for example, has no such loss over time, and is a "permanent" magnet as long as it remains unbroken and superconducting. –  Kevin Reid Jul 15 '11 at 4:11
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