Is the center of a charged spherical shell a point of neutral equilibrium?

[Kolahal Bhattacharya]

Well,I think this is interesting.I invite people to think over it.
consider a charged spherical shell.Throughout its interior,E=0.
Now,consider the centre.From Laplace's equation and Earnshaw's theorem,this point is not a point of potential minimum.So,a charge at this point cannot be in stable equilibrium.Say,you displace it slightly.
What do you find.It just gets stagnant where you left it!
If this is not a point of stable eqlbm,it is not also a point of unstable equilibrium.
So,is it a point of neutral equilibrium?
Apparently seems so.But,I wish to confirm.

 

[Claude Bile]

it is not also a point of unstable equilibrium.

Why not?

Claude.

 

[cesiumfrog]

if you displace it slightly, that is, if you nudge it: it will keep moving in that direction, eventually leaving the interior. Seems unstable.

 

[Ahmes]

While it is the lowest value the potential can have, it doesn't have to count as stable equilibrium because the function is not derivable, and it's not a local minimum.

 

[mheslep]

I'm not sure the premise is set up correctly: If the sphere is _empty_, then yes E=0. But now there's some E once the charge is inserted.

 

[HallsofIvy]

The center is not a point of minimum potential simply because the field is 0 throughout the sphere. Every point in the interior of the sphere is a point of "neutral equilibrium". What's so strange about that?

 

[Kolahal Bhattacharya]

I agree with HallsofIvy:

What's so strange about that?

However,think of the case,where you have a sphere on which charge can move.Say,a conductor.As you move the charge inside,the surface charge distribution may be affected.Then,what would be the conclusion?

 

[StatMechGuy]

I agree with HallsofIvy:

However,think of the case,where you have a sphere on which charge can move.Say,a conductor.As you move the charge inside,the surface charge distribution may be affected.Then,what would be the conclusion?

You would be dealing with an electrodynamics. However, the time it takes for the system to return to the electrostatic limit is extremely short.

 

[Kolahal Bhattacharya]

However,I found it.The resulting case will be unstable equilibrium.