When Rutherford made his discovery, the atom was thought to be a sort of plum pudding, with the positive and negative charge mixed together like the ingredients of a cake. Rutherford conducted an experiment whereby he he shot a beam of alpha particles (Helium nuclei) at layers of gold leaf only a few atoms thick. If the plum-pudding model were correct, one would expect the vast majority of alpha particles to get stuck in the atoms, while some would get through.
The results were intriguing. The majority of alpha particles actually made it through whereas some were deflected by very large angles (over 90o). Rutherford concluded that the majority of the mass was concentrated in a minute, positively charged region (the nucleus) surrounded by electrons. When a (positive) alpha particle approached sufficiently close to the nucleus it was repelled strongly enough to rebound at high angles. The small size of the nucleus explained the small number of alpha particles that were repelled in this way. Rutherford showed, using the method below, that the size of the nucleus was about 10-14 m.
For head on collisions between alpha particles and nucleus', all the kinetic energy (.5mu2) of the alpha particle is turned in to potential energy and the particle is at rest. The distance from the centre of the electron to the centre of the nucleus (b) is approximately then the radius.
Applying the inverse square law can the charges on the electron and nucleus one can write:
For an alpha particle:
Substituting these in gives the value of about 2.7*10-14 m.