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

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Discussion Overview

The discussion revolves around whether the center of a charged spherical shell can be considered a point of neutral equilibrium. It examines concepts related to electrostatics, potential energy, and equilibrium states within the context of a charged spherical shell and the implications of charge displacement.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants note that the electric field inside a charged spherical shell is zero, leading to the suggestion that the center may be a point of neutral equilibrium.
  • One participant argues that displacing a charge from the center would cause it to move away, indicating instability rather than neutrality.
  • Another participant points out that while the potential is at its lowest value at the center, it does not constitute stable equilibrium due to the lack of a local minimum.
  • There is a suggestion that if the sphere is empty, the electric field is zero, but the situation changes once a charge is introduced.
  • One participant claims that every point in the interior of the sphere can be considered a point of neutral equilibrium.
  • Another participant raises a scenario involving a conductor where moving charge could affect surface charge distribution, complicating the equilibrium analysis.
  • One participant concludes that the resulting case with a movable charge would lead to unstable equilibrium.

Areas of Agreement / Disagreement

Participants express differing views on the nature of equilibrium at the center of the charged spherical shell, with some supporting the idea of neutral equilibrium and others arguing for instability. The discussion remains unresolved regarding the classification of equilibrium states.

Contextual Notes

There are limitations in the assumptions made about the nature of the charge distribution and the implications of moving charges within the shell. The discussion also highlights the dependence on whether the shell is considered empty or charged.

Kolahal Bhattacharya
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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.
 
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Kolahal Bhattacharya said:
it is not also a point of unstable equilibrium.

Why not?

Claude.
 
if you displace it slightly, that is, if you nudge it: it will keep moving in that direction, eventually leaving the interior. Seems unstable.
 
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.
 
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.
 
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?
 
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?
 
Kolahal Bhattacharya said:
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.
 
However,I found it.The resulting case will be unstable equilibrium.
 

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