Rutherford Scattering Made Simple

Rutherford scattering experiment1 is so important that it is seldom not mentioned in any introductory modern physics course. For those non-calculus classes, it would be hard to discuss many details, such as the derivation of the differential cross section, about it. Even so, some quantitative discussion on how the scattering angle depends on the impact parameter, based on the classical Coulomb model, will give students a better physical picture. The more complete approach that derive the hyperbolic trajectory by solving the New-ton’s second law, using differential equations2, is obviously not suitable for this purpose. The derivation using the impulse-momentum relation3,4, without knowing the exact form of the trajectory, is neat, but still require integration. This may be hard to teach to a non-calculus class. The method that Rutherford himself and others used1,4 is indeed non-calculus. It assumes the trajectory to be a branch of a hyperbola. Students should not be hard to imagine that the trajectory is hyperbola-like, because of the scattering nature of the problem and that the forces between the particles is repulsive. However, this method also assumes the heavy nucleus to be at the outer focus of the hyperbola. This is not so easy to understand