Equilibrium: stable or unstable, that is the question...

Electricity and Magnetism Level 3

A uniformly charged ring of radius R R with linear charge density λ \lambda lies on the ground. Above it, along its axis, hovers a particle of mass m m and charge q q . The particle is in equilibrium.

Describe the equilibrium of the particle:

(i) If the separation between the charged particle and the centre of the ring is less than R 2 \dfrac{R}{\sqrt{2}}

(ii) If the separation between the charged particle and the centre of the ring is greater than R 2 \dfrac{R}{\sqrt{2}}

Details & Assumptions

  1. The ring retains its charge; it is not lost to the ground.
  2. A uniform gravitational field g ⃗ \vec{g} is acting downward.
(i) Unstable, (ii) Unstable (i) Stable, (ii) Stable Equilibrium is not possible (i) Unstable, (ii) Stable Insufficient information (i) Stable, (ii) Unstable

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Raj Magesh by Raj Magesh , 23, India

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1 solution

For a ring, the maximum Electric Field Intensity is at d = R 2 d = \dfrac{R}{\sqrt{2}} where d is the distance from the centre of the ring.

In case (i), if the particle is slightly pushed forward, it would experience more force, hence unstable.

In case (ii), if the particle is slightly pushed forward, it would experience less force, hence stable.

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