Give the relationship between impact parameter and scattering angle. 

In a Rutherford scattering experiment when a projectile of charge $z_1$ and mass $M_1$ approaches a target nucleus of charge $z_2$ and mass $M_2$, the distance of closest approach is $r_0$ The energy of the projectile is 

Hydrogen $(H)$, deuterium $(D)$, singly ionized helium $(H{e^ + })$ and doubly ionized lithium $(Li)$ all have one electron around the nucleus. Consider $n =2$ to $n = 1 $ transition. The wavelengths of emitted radiations are ${\lambda _1},\;{\lambda _2},\;{\lambda _3}$ and ${\lambda _4}$ respectively. Then approximately

An electron is moving round the nucleus of a hydrogen atom in a circular orbit of radius $r$. The Coulomb force $\vec{F}$ between the two is

In Bohr model of hydrogen atom, the force on the electron depends on the principle quantum number as

$\sqrt{d_{1}}$ and $\sqrt{d_{2}}$ are the impact parameters corresponding to scattering angles $60^{\circ}$ and $90^{\circ}$ respectively, when an $\alpha$ particle is approaching a gold nucleus. For $d_{1}=x d_{2}$, the value of $x$ will be ________