Show the trajectory of $\alpha$-particle for different impact parameters and explain how Rutherford used this to determine the upper limit of the nuclear size.

(N/A) The impact parameter $b$ is the perpendicular distance of the initial velocity vector of the $\alpha$-particle from the center of the nucleus. The trajectory of the $\alpha$-particle depends on this impact parameter $b$.
$A$ given beam of $\alpha$-particles has a distribution of impact parameters $b$,causing the beam to scatter in various directions with different probabilities. In a beam,all particles have nearly the same kinetic energy.
The smaller the impact parameter $b$ of the $\alpha$-particle,the closer it passes to the nucleus,resulting in a larger scattering angle $\theta$.
In the case of a head-on collision,the impact parameter is minimum $(b=0)$ and the $\alpha$-particle rebounds back $(\theta \cong \pi)$.
As the impact parameter $b$ increases,the scattering angle $\theta$ decreases. For very large impact parameters,the $\alpha$-particle continues on its original trajectory without significant scattering $(\theta \cong 0^{\circ})$.
Since only a small fraction of incident particles rebound back,it indicates that the number of $\alpha$-particles undergoing head-on collisions is very small. This implies that the mass and positive charge of the atom are concentrated in a very small volume. Therefore,Rutherford scattering is a powerful method to determine an upper limit to the size of the nucleus.
From this experiment,Rutherford estimated the dimension of the nucleus to be in the range of $10^{-15} \ m$ to $10^{-14} \ m$.