# Ring Trap

Bodies · mechanical systems

```
charges
m=10^6  q=1  x=2  y=0
m=10^6  q=1  x=sqrt(2)  y=sqrt(2)
m=10^6  q=1  x=0  y=2
m=10^6  q=1  x=-sqrt(2)  y=sqrt(2)
m=10^6  q=1  x=-2  y=0
m=10^6  q=1  x=-sqrt(2)  y=-sqrt(2)
m=10^6  q=1  x=0  y=-2
m=10^6  q=1  x=sqrt(2)  y=-sqrt(2)
m=1  q=1  x=0.5  y=0.3  vx=s
```

[Open in the app](https://www.wavelace.com/app#p=151) · [This page](https://www.wavelace.com/presets/ring-trap)

### The setup

Eight charges of +1 sit on a ring of radius 2, one every 45°, each given a mass of a million so that nothing moves them. A ninth charge, also +1 and of unit mass, starts inside at `(0.5, 0.3)` with speed `s` along x. Every charge pushes on every other, so the ring pushes the light one back towards the middle whichever way it goes, and it wanders about inside a cage made of nothing but repulsion.

### The cage

The potential energy of the light charge is the sum of `1/r` over the eight ring charges. It is 4 at the centre, 4.09 where the charge starts, and rises towards the ring, most slowly midway between two ring charges, where it reaches 4.57. That midpoint is the lowest gap in the wall. A charge with more than 0.48 of kinetic energy, `s ≈ 0.99`, has enough to get out through it. Energy is not enough on its own: it must also arrive at a gap heading outwards. At `s = 1.1` the charge had not found a way out after 90 s; at 1.2 it was out through a gap after 29 s.

At the opening value of 0.8 the charge first turns back after 1.3 s and never gets further than 1.15 from the centre, well short of the ring at 2. Earnshaw's theorem says no arrangement of fixed charges can hold another still, and it does not: the trapped charge never stops, it is only kept.

### Try

- Drag `s` up past 1.2. Above the escape energy the charge finds a gap sooner or later; at 1.2 it takes 29 s, at 2 it is gone in the first second.
- Drag `s` down to 0.4: the charge keeps between 0.2 and 0.76 of the centre and rattles in a small, ragged loop.
- Flip the light charge, `m=1  q=-1` on the last line. Unlike charges attract, so it falls straight onto the nearest ring charge, and the energy drift readout jumps as the two meet.
- `Top` in the deck shows the ring and the path inside it flat.

### Read more

- [Earnshaw's theorem](https://en.wikipedia.org/wiki/Earnshaw%27s_theorem)
- [Coulomb's law](https://en.wikipedia.org/wiki/Coulomb%27s_law)
- [Electric potential energy](https://en.wikipedia.org/wiki/Electric_potential_energy)