Explanation
Informally, the case of a point charge in an arbitrary static electric field is a simple consequence of Gauss's law. For a particle to be in a stable equilibrium, small perturbations ("pushes") on the particle in any direction should not break the equilibrium; the particle should "fall back" to its previous position. This means that the force field lines around the particle's equilibrium position should all point inwards, towards that position. If all of the surrounding field lines point towards the equilibrium point, then the divergence of the field at that point must be negative (i.e. that point acts as a sink). However, Gauss's Law says that the divergence of any possible electric force field is zero in free space. In mathematical notation, an electrical force deriving from a potential will always be divergenceless (satisfy Laplace's equation):
Therefore, there are no local minima or maxima of the field potential in free space, only saddle points. A stable equilibrium of the particle cannot exist and there must be an instability in at least one direction.
To be completely rigorous, strictly speaking, the existence of a stable point does not require that all neighboring force vectors point exactly toward the stable point; the force vectors could spiral in towards the stable point, for example. One method for dealing with this invokes the fact that, in addition to the divergence, the curl of any electric field in free space is also zero (in the absence of any magnetic currents).
This theorem also states that there is no possible static configuration of ferromagnets which can stably levitate an object against gravity, even when the magnetic forces are stronger than the gravitational forces.
Earnshaw's theorem has even been proven for the general case of extended bodies, and this is so even if they are flexible and conducting, provided they are not diamagnetic, as diamagnetism constitutes a (small) repulsive force, but no attraction.
There are, however, several exceptions to the rule's assumptions which allow magnetic levitation.
Read more about this topic: Earnshaw's Theorem
Famous quotes containing the word explanation:
“The explanation of the propensity of the English people to portrait painting is to be found in their relish for a Fact. Let a man do the grandest things, fight the greatest battles, or be distinguished by the most brilliant personal heroism, yet the English people would prefer his portrait to a painting of the great deed. The likeness they can judge of; his existence is a Fact. But the truth of the picture of his deeds they cannot judge of, for they have no imagination.”
—Benjamin Haydon (1786–1846)
“There is no explanation for evil. It must be looked upon as a necessary part of the order of the universe. To ignore it is childish, to bewail it senseless.”
—W. Somerset Maugham (1874–1965)
“Herein is the explanation of the analogies, which exist in all the arts. They are the re-appearance of one mind, working in many materials to many temporary ends. Raphael paints wisdom, Handel sings it, Phidias carves it, Shakspeare writes it, Wren builds it, Columbus sails it, Luther preaches it, Washington arms it, Watt mechanizes it. Painting was called “silent poetry,” and poetry “speaking painting.” The laws of each art are convertible into the laws of every other.”
—Ralph Waldo Emerson (1803–1882)