# Earnshaw’s Theorem and Ring of Charge

A classic problem in determining the motion of a negative charge when displaced from a positively charged ring shows that the charge oscillates.

However, Earnshaw’s theorem states that (quoting Griffiths) ‘A charged particle cannot be held in stable equillibrium by electrostatic forces alone’. However, the system above seems to be stable. What causes this seemingly contradictory results?

• If you displace the particle along the axis that passes through the center of the ring and perpendicular to the plane of the ring, you get a restoring force. However, it seems likely that if you displace the particle instead in the plane of the ring towards one part of the ring, there is no restoring force; instead, there's a force pointed toward the nearest part of the ring. That's my guess. – march Mar 24 at 2:59
• Earnshaw's Theorem according to wikipedia is applicable on point charges. However even in this case, why do you think that the ring will remain stationary? You will require a force to hold it in place. – harshit54 Mar 24 at 3:43
• Also read this and this. – harshit54 Mar 24 at 3:44

Displacement along the symmetry axis results in a restoring force along the symmetry axis as you have calculated.