Can a Positive Charge be in Stable Equilibrium in the Center of a Square According to Earnshaw's Theorem?

[lamp post]

we have got four charges placed at the ends of a square. if we palce a positive charge in the center, it appeares to be in stable equillibrium; yet earnshaw's theorem(that no particle can be in stable equillibrium if it is under the action of electrostatic forces only) holds. Why is that?

 

[Chandra Prayaga]

Earnshaw's theorem

The charge at the center of the square is in equilibrium, yes. But it is not in stable equilibrium. Stable equilibrium means that if you move the charge slightly in any direction, then it should return to the center. In this case, as in every other case involving electrostatic forces only, if you move the charge, from the center, it will not return. Check that out.

 

[GSXR750]

Earnshaw's theorem states that no particle can be in stable equilibrium solely under the influence of electrostatic forces. This means that in the scenario described, the positive charge in the center of the square may appear to be in stable equilibrium, but it is actually not. This is because the electrostatic forces acting on the positive charge from the four charges at the ends of the square are not balanced and will eventually cause the charge to move.

This can be explained by considering the forces acting on the positive charge. Since the four charges at the ends of the square are all positive, they will repel the positive charge in the center. However, the forces acting from the corners of the square are not equal in magnitude or direction. This means that the electrostatic forces are unbalanced and will result in the positive charge moving away from the center.

Earnshaw's theorem is a fundamental principle in electrostatics and it applies to all systems of charged particles. It is a consequence of the inverse square law, which states that the force between two charged particles is inversely proportional to the square of the distance between them. This means that the force decreases rapidly as the distance between the particles increases, making it impossible for particles to be in stable equilibrium solely under the influence of electrostatic forces.

In summary, the apparent stable equilibrium of the positive charge in the center of the square is an illusion caused by the unbalanced electrostatic forces acting on it. Earnshaw's theorem holds true in this scenario because the electrostatic forces are not balanced and will eventually cause the positive charge to move.