The discussion centers around Earnshaw's Theorem, which posits that it is impossible to achieve stable levitation of a charged object using static electric fields. Participants seek to understand the implications of this theorem, particularly in relation to Gauss's Law and the behavior of electric fields at equilibrium points.
Participants do not reach a consensus on the implications of Earnshaw's Theorem or its relationship to Gauss's Law. Multiple competing views and interpretations remain, particularly regarding the nature of stability and the behavior of electric fields.
Some participants express uncertainty about the application of Gauss's Law in the context of Earnshaw's Theorem, particularly regarding the implications of electric field lines and enclosed charge. There are unresolved questions about the mathematical relationships involved in these concepts.
Gauss's Law says that the divergence of any possible electric force field is zero in free space.
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.
kcdodd said:It is not that it is impossible to levitate an object. It is just impossible to make the levitation stable. An object in equilibrium will be unstable in at least one direction. A small displacement from equilibrium will grow and take the object further away. For instance, it is easy to see how an object can be stable in the vertical direction. If it is displaced up then the electrostatic force decreases and gravity will bring it down. Displace it down and electric force increases and pushes it up. But, if you displace it horizontally then the force points in the direction displaced and will push the object further in that direction, not back.
If it is displaced up then shouldn't the electrostatic force increase the closer the charged object comes to the source of electric field?
E=V/d=F/q
therefore
F=V*q/d
?
Or am I being stupid
Now, you can create a combination of charges to create a saddle point, and place the object so that it is stable in the horizontal direction. However, this loses vertical stability. If the object is displaced vertically in the saddle point then the electric force actually increases now, instead of decreases, which means it will move further away from equilibrium.
Why will it now reverse what is the reason??
I am guessing you are describing a penning trap now?
kcdodd said:The electric field cannot point toward the null from all directions as that would constitute net flux, and Q_enc != 0.
kcdodd said:If it points toward the null in all directions then there must be non-zero flux through the gaussian surface, which means Q is non-zero.
0xDEADBEEF said:What is so hard to understand?
So if you take a small volume around the charge and apply Gauss' law to the surface you find that another opposite charge must be trapped inside this volume with the first one, for arbitrarily small volumes.
0xDEADBEEF said:Hence you cannot get a minimum without a charge in it, so you cannot levitate an electron without it falling onto a proton, we need quantum mechanics for that.