Problem 2P

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 corner. 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.]

Step 1 of 1

We have to justify Earnshaw’s Theorem in one sentence by considering the cubical arrangement of fixed charges as given in the figure.

A stable equilibrium is a point of local minimum in the potential energy. Now, here the potential energy is qV. But Laplace’s equation does not allow any local minima for V. So what looks like a minimum, in the cubical arrangement of fixed charges must in fact be a saddle point.

The leak in the box is through the centre of each face of the cube.