Example · Rutherford Scattering
Problem statement
Simulate an alpha particle scattering off a gold nucleus and verify the classic relation between impact parameter \(b\) and scattering angle \(\theta\):
\[\tan\left(\frac{\theta}{2}\right) = \frac{q_1 q_2}{4\pi\varepsilon_0\,\mu\, b\, v_\infty^2}\]
Given information
- Coulomb field of the nucleus: \(\mathbf{E} = \dfrac{q_2}{4\pi\varepsilon_0 r^2}\hat{\mathbf{r}}\) — the first non-uniform field of the course
- Alpha particle arrives from far away with speed \(v_\infty\) and perpendicular offset (impact parameter) \(b\); \(\mu\) = reduced mass
Solution strategy
- Write the acceleration \(\mathbf{a}(\mathbf{x}) = \frac{q_1}{m}\mathbf{E}(\mathbf{x})\) as a state-space function.
- Integrate with RK4 (accuracy matters near closest approach; no long-time issues here).
- Start far upstream (\(r \gg b\)); run until far downstream; measure \(\theta = \arctan(v_y/v_x)\big|_\text{out}\).
- Sweep \(b\) over a range and plot \(\theta(b)\) against the formula.
Result
The simulated points fall on the theoretical hyperbola-scattering curve across the sweep — small \(b\) → near-backscatter (\(\theta \to 180°\)), large \(b\) → gentle deflection. The historic conclusion, re-derived numerically: only a concentrated positive charge (a nucleus) produces large-angle scattering; Thomson's smeared "pudding" cannot.
Key takeaways
- The step from uniform to \(1/r^2\) fields costs nothing in the code — the field function is just swapped (the whole design of the course solver).
- Energy conservation is the run's health check: the Coulomb force is conservative, so \(\frac12\mu v^2 + \frac{q_1q_2}{4\pi\varepsilon_0 r}\) must return to its initial value downstream.
- Watch the step size near perihelion: fast force variation is where fixed-step integrators bleed accuracy (motivation for adaptive stepping — and for softening when many bodies are involved).