Question

If you wanted to use the idealized trap of Fig. 39-1 to trap a positron, would you need to change

(a) the geometry of the trap,

(b) the electric potential of the central cylinder, or

(c) the electric potentials of the two semi-infinite end cylinders?

(A positron has the same mass as an electron but is positively charged.)

Short answer

(a) The geometry of the trap need not be changed to trap a positron.

(b) The potential of the central cylinder need not be changed to trap a positron.

(c) The potential of the two semi-infinite end cylinders need to changed to $+\infty$to trap the positron.

Step-by-step solution

Given data:

A positron is required to be trapped in a three-cylinder setup out of which the central cylinder is of finite length and is at the center and the other two are semi-infinitely long and are on both sides of the center cylinder.

Potential energy:

The potential energy U of a particle of charge q in a region of potential V is

$\mathit{U}\mathbf{=}\mathit{q}\mathit{V}$ ..... (I)

In the electron trap, the central cylinder is kept at zero potential and the semi-infinitely long cylinders are kept at infinitely negative potential.

(a) Determining whether the geometry of the trap needs to be changed to trap a positron:

The geometry of the trap doesn't take any useful part in the trap except for the fact that the left and right cylinders have to be infinitely long so that almost everywhere outside the central cylinder, the potential is constant. This has to be the case for the positron too. Hence the geometry need not be changed.

(b) Determining whether the potential of the central cylinder needs to be changed to trap a positron:

In the electron trap, the central cylinder is kept at zero potential so that from equation (I), the electron has zero potential energy inside it. The charge of the positron is positive but in a zero potential region, its potential energy will also be zero. Hence the potential of the central cylinder need not be changed.

(c) Determining whether the potential of the semi-infinite cylinders needs to be changed to trap a positron:

If the semi-infinite end cylinders are kept at potential $-\infty$, the potential energy of the positron, from equation (I) will also be $-\infty$outside the central cylinder. So if it moves to the sides, an infinite force will pull it out. The potential of the side cylinders can be made $+\infty$.

As the positron is of the same mass of the electron, there is no need to change the geometry of the trap. Also, the central potential is at zero, so there is no need to change the central potential.

But as the positron has a positive charge, if you want to trap it, the potentials of the two walls of the well must be positive. That is the electric potentials of the two semi-infinite end cylinders must be changed.

Hence, the setup will act as an infinite potential well and the positron will be forced to stay inside the central cylinder.