Earnshaw's Theorem: Stability in Electrostatics?

[kini.Amith]

In Electrodynamics text by Griffiths there is the statement of Earnshaw's theorem "a charged particle cannot be held in a stable equilibrium by electrostatic forces alone." But if we consider the system in which a positive charge is placed midway(where E is zero) between two positive charges of equal magnitude which are held in position by external forces. If the charge in the middle is displaced axially , then the electrostatic force will force it back into the equilibrium position.So isn't the charge in stable equilibrium. Isn't this a violation of Earnshaw's theorem?

 

[Boorglar]

In order for the equilibrium to be stable, it must force back any small displacement from the equilibrium, in every direction, not just axially.
Consider what happens when you move the charge in a perpendicular direction to the axis.

 

[zoki85]

In Electrodynamics text by Griffiths there is the statement of Earnshaw's theorem "a charged particle cannot be held in a stable equilibrium by electrostatic forces alone." But if we consider the system in which a positive charge is placed midway(where E is zero) between two positive charges of equal magnitude which are held in position by external forces. If the charge in the middle is displaced axially , then the electrostatic force will force it back into the equilibrium position.So isn't the charge in stable equilibrium. Isn't this a violation of Earnshaw's theorem?

Collection of charged particles, not "a charged particle". See: http://en.wikipedia.org/wiki/Earnshaw%27s_theorem

 

[kini.Amith]

Collection of charged particles, not "a charged particle". See: http://en.wikipedia.org/wiki/Earnshaw%27s_theorem

Is this true? i have seen the wikipedia page, but the text specifically says"a charged particle". Is it not valid for a single particle?

In order for the equilibrium to be stable, it must force back any small displacement from the equilibrium, in every direction, not just axially.
Consider what happens when you move the charge in a perpendicular direction to the axis.

I see. So it is valid only in three dimensions.Thanks

 

[Nabeshin]

Is this true? i have seen the wikipedia page, but the text specifically says"a charged particle". Is it not valid for a single particle?

I guess the reason for the caveat here may be that a single charged particle is clearly in equilibrium.

 

[A.T.]

I guess the reason for the caveat here may be that a single charged particle is clearly in equilibrium.

But not in a stable equilibrium as stated in the theorem, just in a neutral equilibrium.