Earnshaw's Theorem and charges

  • Context: Graduate 
  • Thread starter Thread starter pivoxa15
  • Start date Start date
  • Tags Tags
    Charges Theorem
Click For Summary

Discussion Overview

The discussion revolves around Earnshaw's Theorem and its implications regarding the trapping of charged particles in electrostatic fields. Participants explore the conditions under which a charged particle can be considered trapped, the nature of equilibrium points, and the role of charge distributions in determining stability.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Exploratory

Main Points Raised

  • Some participants interpret Earnshaw's Theorem as stating that a charged particle cannot be trapped in a charge-free electrostatic field, suggesting that any region capable of trapping a charge must contain other charges.
  • Others argue that local minima of potential, which indicate stable equilibrium, can only occur where charge density is non-zero, while local maxima (unstable equilibrium) can exist where charge density is zero.
  • A participant illustrates that in a system with two equal negative charges, the midpoint is an unstable equilibrium for both positive and negative charges, while the positions of the charges themselves are stable for a positive charge.
  • Some participants challenge the notion that a charge can be trapped in a vacuum, arguing that without an energy minimum, the particle cannot be considered truly trapped.
  • There is a discussion about the implications of a true vacuum and whether a particle with zero initial kinetic energy could be trapped, with some asserting that without potential, there is no confining force to create a local minimum.
  • One participant emphasizes that even in a vacuum, an infinitesimal nudge would cause the particle to move, reinforcing the idea that Earnshaw's Theorem pertains specifically to stable equilibrium.

Areas of Agreement / Disagreement

Participants express differing interpretations of Earnshaw's Theorem and its implications, with no consensus reached on the conditions necessary for trapping a charge or the nature of equilibrium points. The discussion remains unresolved regarding the validity of certain claims and the implications of specific scenarios.

Contextual Notes

Participants highlight limitations in their arguments, such as the dependence on definitions of equilibrium and the assumptions regarding charge distributions. The discussion also reflects uncertainty about the implications of a vacuum and the nature of potential energy in trapping scenarios.

pivoxa15
Messages
2,250
Reaction score
1
The theorem states:
"You can't trap a charged particle in a charge-free electrostatic field; there is no point of stable equilibrium.'
R. Good, p284.

Is this theorem trying to say that if you are ever able to trap a charge x inside any region than that region must already contain a charge y (although that y was not trapped in the region beforehand).

I suppose any region of interest must have electric fields through it so any charge placed inside it will have a force on it but with divF=0. Hence the froces cannot be directed in a way so as to trap the particle.

What about a region without any electric fields than if the charge is placed inside it with 0 intitial velocity than it will stay stationary. But a region like this does not apply to the theorem because it assumes an electrostatic field, not a vacuum.
 
Last edited:
Physics news on Phys.org
pivoxa15 said:
The theorem states:
"You can't trap a charged particle in a charge-free electrostatic field; there is no point of stable equilibrium.'
R. Good, p284.

Is this theorem trying to say that if you are ever able to trap a charge x inside any region than that region must already contain a charge y (although that y was not trapped in the region beforehand).
I believe what it is saying is that any distribution of charges will result in local minima of the potential (ie, points of stable equilibrium) only at points where the charge density is non-zero. Local maxima (which are points of unstable equilibrium) however, can exist at points where the charge density is zero.

For instance, if you have two equal negative charges separated by some distance, the midpoint of the line joining the charges is a point of unstable equilibrium for both positive and negative charges placed there. However, the positions of the charges themselves are points of stable equilibrium for a positive charge. If you take a positive charge and place it at the midpoint, it will stay there so long as there are no perturbations. The slightest distaurbance will likely cause it to collapse into one of the two neegative charges.
 
Last edited:
Gokul43201 said:
For instance, if you have two equal negative charges separated by some distance, the midpoint of the line joining the charges is a point of unstable equilibrium for both positive and negative charges placed there. However, the positions of the charges themselves are points of stable equilibrium for a positive charge. If you take a positive charge and place it at the midpoint, it will stay there so long as there are no perturbations. The slightest distaurbance will likely cause it to collapse into one of the two neegative charges.

This is the case of unstable equilibrium so the theorem does not apply.

What do you think of the validity of
'This theorem is trying to say that if you are ever able to trap a charge x inside any region than that region must already contain a charge y.' where x and y are nonzero?
 
pivoxa15 said:
This is the case of unstable equilibrium so the theorem does not apply.
That's not true. For the described situation, there exists one point of unstable equilibrium, and two points of stable equilibrium (ignoring points at infinity). The goal of the illustration was to demonstrate that, in accordance with the theorem, the only point of equilibrium where the charge density is zero is not a stable one.

What do you think of the validity of
'This theorem is trying to say that if you are ever able to trap a charge x inside any region than that region must already contain a charge y.' where x and y are nonzero?
Yes, that works.
 
Gokul43201 said:
That's not true. For the described situation, there exists one point of unstable equilibrium, and two points of stable equilibrium (ignoring points at infinity). The goal of the illustration was to demonstrate that, in accordance with the theorem, the only point of equilibrium where the charge density is zero is not a stable one.

For this situation to have a charge density 0, it means the mid charge must contain twice the charge of one of the end charges.

i.e

(-1C) (+2C) (-1C)

C=coulomb

From the electrostatics of the situtation, its true that the midcharge will be in unstable equilbrium but the two end charges will not be in any equilibirum and wiill be attracted toward the centre positive charge.
Even without the centre positive charge, how can the two negative charges be in equillibrium in the first place because they will move to get as far from each other as possible. Or are you assuming they are bound somehow such as being inside a container?
 
pivoxa15 said:
What about a region without any electric fields than if the charge is placed inside it with 0 intitial velocity than it will stay stationary. But a region like this does not apply to the theorem because it assumes an electrostatic field, not a vacuum.

This would not work anyway, because technically you are not trapping the particle. Some energy minimum must exist in order for the particle to exist in a true 'trapped' state.

As Gokul as pointed out, it is not enough that it exist in an unstable equilibrium, because the particle is not trapped any more that it would be sitting in a vacuum at infinity because the slightest imbalance would cause it to accelerate away from its initial position.

Claude.
 
Claude Bile said:
This would not work anyway, because technically you are not trapping the particle. Some energy minimum must exist in order for the particle to exist in a true 'trapped' state.

Claude.

If we imagined a true vacuum existed than everywhere has a minimum energy. If we give the particle no initial kinetic energy than would it be trapped?
 
pivoxa15 said:
If we imagined a true vacuum existed than everywhere has a minimum energy. If we give the particle no initial kinetic energy than would it be trapped?

If there's no potential, then there's no confining potential, and so you can't have a local minimum or whatnot because there's nothing there. It won't move, but what you've just described is about the most uninteresting problem in physics.

Having zero potential everywhere just means that the kinetic energy and momentum are both conserved, and there is nothing to study at this point.
 
pivoxa15 said:
If we imagined a true vacuum existed than everywhere has a minimum energy. If we give the particle no initial kinetic energy than would it be trapped?

No, because if you gave it an infintesimal nudge, it would begin moving.

Claude.
 
  • #10
Claude Bile said:
No, because if you gave it an infintesimal nudge, it would begin moving.

Claude.

That is a good point because Earnshaw's theorem only concerns stable equilibrium.

There is still the question in post 5 to be settled.
 

Similar threads

Undergrad Can nested solenoids trap particles?
  • · Replies 12 ·
Replies
12
Views
1K
High School Can charges move without a field? How charges redistribute without it?
  • · Replies 16 ·
Replies
16
Views
2K
Undergrad Helmholtz's theorem and charge density
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Where to find this uniqueness theorem of electrostatics?
  • · Replies 27 ·
Replies
27
Views
3K
Graduate Ponderomotive force and conservation of momentum
  • · Replies 1 ·
Replies
1
Views
1K
High School Definition of ground potential in infinite sheet charge distributions
  • · Replies 11 ·
Replies
11
Views
2K
Graduate Earnshaw's Theorem: Stability in Electrostatics?
  • · Replies 5 ·
Replies
5
Views
3K
Graduate Electric field of infinite sheet with time dependent charge
  • · Replies 22 ·
Replies
22
Views
2K
Undergrad Conditions for applying Gauss' Law
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Exploring the Electric Field of a Moving Charged Spherical Shell
  • · Replies 14 ·
Replies
14
Views
3K