Question

In one sentence, justify Earnshaw's Theorem: A charged particle cannot be held in a stable equilibrium by electrostatic forces alone. As an example, consider the cubical arrangement of fixed charges in Fig. 3.4. It looks, off hand, as though a positive charge at the center would be suspended in midair, since it is repelled away from each comer. Where is the leak in this "electrostatic bottle"? [To harness nuclear fusion as a practical energy source it is necessary to heat a plasma (soup of charged particles) to fantastic temperatures-so hot that contact would vaporize any ordinary pot. Earnshaw's theorem says that electrostatic containment is also out of the question. Fortunately, it is possible to confine a hot plasma magnetically.]

Short answer

The cube leaks from the centre of each surface.

Step-by-step solution

Stable equilibrium of particle.

If the particle is disturbed by the internal or external force, which tends to the displacement of particle and the particle comes again in the equilibrium, it is known as the stable equilibrium. The particle can have instability only in one order.

Determine the leaks in the cubic arrangement.

The saddle point in the free space is defined as the stationary point with a non-local extremum. The saddle point in free space has an increasing curve in one direction and a decreasing curve in the other.

For the given arrangement, the potential is contour increasing along the diagonal and decreasing along the x and y-axis. Therefore, the potential due to the positive charge will rise in one direction and drop in another.

Hence the cube leaks from the centre of each surface.