# The trajectory of charged particles projected with finite impact parameter

Actually, I wanted a tool that simulates the trajectory of charged particles while they interact with each other in the presence of their electric fields.

I wanted the simulation for a particular problem which needs a positively charged particle to be thrown from a large distance (infinite distance) towards another 'fixed' positively charged particle with a finite impact parameter, and thus find the angle of deviation of the thrown particle from its initial direction of motion when their distance of approach is minimum.

I would appreciate if someone tells me a way to find this angle using only fundamentals of physics i.e. without plotting its trajectory.

The minimum distance of approach can be found out precisely be conserving the angular momentum about the fixed charge and also conserving the total mechanical energy of the system.

$$\Theta=2\arctan{\frac{q_1q_2}{4\pi\epsilon_0mv_0^2b}},$$
where $$m$$ is the mass of the moving charge, $$v_0$$ is the speed of the moving charge at infinity and $$b$$ is the impact parameter.
• Also note that due to time inversion symmetry, the angle at closest approach (asked for) is half the Rutherford scattering angle. Derivations generally rotate the trajectory to exemplify its symmetry about some $y$-axis, define the angle between this axis and the line between the particles as some $\phi$, and then time-integrate the force in the $y$-direction; a denominator is just the angular momentum which is constant. (See e.g. the derivation on Hyperphysics which gets there eventually with a cotangent expression.) – CR Drost Jul 14 '19 at 18:45