# Rutherford Scattering

Bodies · mechanical systems

```
charges
m=1000  q=1  x=0  y=0
m=1  q=1  x=-3  y=b  vx=2
```

[Open in the app](https://www.wavelace.com/app#p=150) · [This page](https://www.wavelace.com/presets/rutherford-scattering)

### The setup

A heavy charge of mass 1000 sits at the origin. A unit charge of the same sign is fired from `(−3, b)` at speed 2, straight along x, so it would pass the origin at a distance `b` if nothing pushed it. Both charges are +1, so Coulomb's law with `k = 1` pushes them apart, and the light one bends away from the centre on a hyperbola while the heavy one barely moves. Every gravity preset bends towards its centre; this is the other sign.

### The angle

The slider `b` is the impact parameter, and dragging it fires the shot again from the start. Rutherford's formula gives the deflection as `2·atan(qQ / 2Eb)` with the kinetic energy `E = ½·1·2² = 2` and `qQ = 1`, so at `b = 0.5` a shot from infinity would turn through 53°. Here it turns through 49°, because it starts only 3 away with a third of a unit of potential energy already in hand. At `b = 0.2` the measured angle is 98° against 103°, at `b = 1` it is 25° against 28°: the closer the aim, the sharper the turn.

Head on, at `b = 0`, the charge slows, stops and comes straight back. It stops at 0.43 from the centre, where its whole energy of 2.33 has become potential energy `1/r`.

### History

Hans Geiger and Ernest Marsden fired alpha particles at gold foil in 1909 and saw a few come straight back, which the atom of the day could not do. Ernest Rutherford explained it in 1911 with exactly this picture, a small heavy charge at the centre of the atom scattering the alphas on hyperbolae, and derived the angle formula above. The nucleus was born of that calculation.

### Try

- Drag `b` towards 0 and watch the hyperbola tighten; at 0 the charge turns straight back at 0.43 from the centre.
- Flip the sign, `q=-1` on the second line. Unlike charges attract, and the same shot now bends towards the centre and past it, through 57° the other way, coming within 0.32 of the heavy charge.
- Lighten the target, `m=4` on the first line, an alpha particle on helium. The light charge still turns through 49° but the target recoils, moving 2.6 in the same time at a speed of 0.4.
- `Top` in the deck shows the hyperbola flat, with the heavy charge at its focus.

### Read more

- [Rutherford scattering](https://en.wikipedia.org/wiki/Rutherford_scattering)
- [Geiger–Marsden experiments](https://en.wikipedia.org/wiki/Geiger%E2%80%93Marsden_experiments)
- [Coulomb's law](https://en.wikipedia.org/wiki/Coulomb%27s_law)
- [Impact parameter](https://en.wikipedia.org/wiki/Impact_parameter)