For HL students: calculate the distance of closest approach by equating the projectile's initial kinetic energy to its maximum electric potential energy. For a head-on approach from a very large distance, , so .
For HL students: calculate the distance of closest approach by equating the projectile's initial kinetic energy to its maximum electric potential energy. For a head-on approach from a very large distance, , so .
This calculation applies conservation of energy to a positively charged projectile, such as an alpha particle, approaching a positively charged nucleus. The repulsive electrostatic force reduces the projectile's speed and converts kinetic energy into electric potential energy. This is Additional Higher Level syllabus point E.1.9.
At the distance of closest approach, the projectile is momentarily stationary, so its kinetic energy is zero. If it starts effectively at infinity, its initial electric potential energy is also zero. Therefore:
For an alpha particle, . For a nucleus with proton number , , giving:
This result is valid only for zero impact parameter. If the projectile has angular momentum, it does not stop completely at its nearest point, because tangential kinetic energy remains; a different energy treatment is required.
| Assumption | Why it matters |
|---|---|
| Head-on approach | The projectile moves directly toward the nucleus and reaches a turning point. |
| Nucleus remains stationary | Nuclear recoil energy is neglected. |
| Electrostatic interaction only | The model applies before the short-range strong nuclear force becomes significant. |
A alpha particle approaches a gold nucleus with . First convert the kinetic energy:
Then substitute :
The value is about 45 fm, allowing comparison with the approximate size of a gold nucleus. Since it lies outside the nucleus, the Coulomb model is reasonable at the turning point.
A common misconception is that this distance must equal the nuclear radius. It is actually the electrostatic turning point and may lie outside the nuclear surface.
In a Calculate question, show the energy conversion, charge substitution, rearrangement, and final answer in metres. State the assumptions if the question asks you to explain the model.