Wavelace

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 The dials and keys named below are the app's.

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.

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Bodies · mechanical systems