The Rutherford experiment successfully demonstrated the...

The Rutherford experiment successfully demonstrated the existence of the nucleus and put limits on the size of the nucleus from the scattering of alpha particles from gold foils. Assume the alpha particle was fired with a speed of about $5.00 \%$ of the speed of light. a) Derive the upper bound of the radius of nucleus in terms of speed of the alpha particle that is scattered in the backward direction. b) Calculate the approximate radius of the gold nucleus with this information.

The Rutherford experiment successfully demonstrated the existence of the nucleus and put limits on the size of the nucleus from the scattering of alpha particles from gold foils. Assume the alpha particle was fired with a speed of about $5.00 \%$ of the speed of light.
a) Derive the upper bound of the radius of nucleus in terms of speed of the alpha particle that is scattered in the backward direction.
b) Calculate the approximate radius of the gold nucleus with this information.

Key Concepts

Energy Conservation in Scattering

Energy conservation is a key principle used to analyze the interactions between the alpha particle and the nucleus. By equating the kinetic energy of the alpha particle with the Coulomb potential energy at the point of closest approach, it is possible to derive an expression for the minimal distance of approach. This condition, particularly when considering backscattered particles, serves as an upper bound on the size of the nucleus.

Rutherford Scattering Experiment

This concept pertains to the experimental observations that led to the discovery of the atomic nucleus. The experiment involved directing positively charged alpha particles at a thin foil and observing their deflected paths. Analysis of the scattering angles provided key insights into the concentration of positive charge within a very small central region, ultimately revealing the existence of a dense nucleus and laying the groundwork for modern atomic theory.

Coulomb Potential Energy

This concept involves the electrostatic potential energy between two charged particles as described by Coulomb's law. In the context of scattering experiments, the repulsive Coulomb force between the positively charged alpha particle and the nucleus plays a vital role. The conversion of kinetic energy to electrical potential energy during the close approach determines how far the alpha particle can get, thereby setting a limit on the nuclear radius.

Estimation of Nuclear Size

This concept involves using the kinematic data from scattering experiments, such as the speed of the incoming alpha particles and the angles of scattering, to establish limits on the dimensions of the nucleus. By employing the relationship between kinetic and potential energies, one derives an estimate for the nuclear radius. This estimation process reveals that the nucleus is extremely small compared to the overall size of the atom, a result that has significant implications for nuclear and atomic physics.

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