Earnshaw's Theorem: Stability in Electrostatics?

• kini.Amith
In summary, the theorem is not violated when a single charged particle is in equilibrium, just in a neutral equilibrium.
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?

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.

kini.Amith said:
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

zoki85 said:
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?
Boorglar said:
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

kini.Amith said:
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.

Nabeshin said:
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.

1. What is Earnshaw's Theorem?

Earnshaw's Theorem is a fundamental principle in electrostatics that states that a system of point charges cannot be held in stable equilibrium solely by the electrostatic interaction between the charges.

2. Why is Earnshaw's Theorem important?

Earnshaw's Theorem has important implications in fields such as physics, chemistry, and engineering. It helps us understand the behavior of charged particles and the stability of systems in electrostatics. It also plays a crucial role in the design of electronic devices.

3. Can Earnshaw's Theorem be violated?

No, Earnshaw's Theorem is a proven mathematical principle and has been rigorously tested and verified. It has not been violated in any known physical system.

4. How does Earnshaw's Theorem relate to the stability of atoms and molecules?

Earnshaw's Theorem explains why atoms and molecules are stable. The negatively charged electrons are attracted to the positively charged nucleus, while the repulsive forces between the positively charged protons are balanced by the attractive forces of the electrons. This balance of forces allows atoms and molecules to exist in stable equilibrium.

5. Are there any exceptions to Earnshaw's Theorem?

Earnshaw's Theorem applies to systems of point charges, but there are exceptions when considering extended charged objects or systems with non-point charges. In these cases, other factors such as shape, size, and external forces may play a role in the stability of the system.

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