E.1.9 (HL)—Closest approach

Use energy conservation

For a head-on alpha particle, the initial kinetic energy is converted into electric potential energy as the particle approaches the positive nucleus. At the turning point, the radial kinetic energy is zero.

Set the energies equal

At the turning point the radial kinetic energy is zero, so the initial kinetic energy equals the electric potential energy of the repulsive alpha-particle–nucleus system. Both positive charges must be included.

Worked example — alpha particle toward gold

For $E_k=5.0\,\mathrm{MeV}=8.0\times10^{-13}\,\mathrm{J}$, $q_\alpha=2e$ and $q_N=79e$, $r_{\min}=k(2e)(79e)/E_k=4.5\times10^{-14}\,\mathrm{m}$. This is a turning-point distance, not automatically the nuclear radius.

Check the turning point

At closest approach the alpha particle has momentarily stopped moving toward the nucleus, then reverses. A larger initial kinetic energy gives a smaller closest-approach distance.

Common trap

Do not use the charge of gold alone: the interaction contains both $q_\alpha$ and $q_{nucleus}$. Also do not leave energy in MeV while using $k$ in SI units.