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Earnshaw's theorem
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Charges cannot be held in stable equilibrium by electrostatic forces alone.
No collection of static point charges can be held in stable equilibrium by electrostatic forces alone (Laplace's equation has no minima). Stability needs other forces, such as motion or quantum effects.
Atoms are stable thanks to quantum mechanics.
Fails with dynamic fields (ion traps).
In more detail
Earnshaw's theorem states that a collection of point charges cannot be held in stable equilibrium by electrostatic forces alone. This is why levitation needs other forces, such as magnetic fields or motion.
Real-world uses
- Magnetic levitation trains use magnetic fields to overcome Earnshaw's theorem.
- Ion traps use oscillating fields to hold charged particles.
- Physicists use Earnshaw's theorem to explain why a charged particle cannot be trapped in a stable point by static fields alone.
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Frequently asked questions
Can electrostatic traps exist at all?
Yes, with time-varying fields or additional forces, as in Paul traps.
What does stable mean here?
That a small displacement produces a restoring force in every direction.
How do Penning and Paul traps then work?
They use changing fields or magnetic fields to create a stable trap, which Earnshaw's theorem does not forbid.
Related topics in Electrostatics
Multipole expansion · Boundary conditions at interfaces · Electrostatic energy of a sphere · Electric field of a ring · Electric field of a disc