Prove that, in Bohr model of hydrogen atom, the time period of revolution of an electron in $\displaystyle \mathrm{n}^{\text {th }}$ orbit is proportional to $\displaystyle \mathrm{n}^{3}$.

$\displaystyle T=\dfrac{2\pi r}{v}$ ---------- ($\displaystyle 1$)From Bohr's quantization condition$\displaystyle mvr=\dfrac{nh}{2\pi}$$\displaystyle v=\dfrac{nh}{2\pi mr}$ ---------- ($\displaystyle 2$)From ($\displaystyle 1$) and ($\displaystyle 2$)$\displaystyle T=\dfrac{2\pi r}{\left(\dfrac{nh}{2\pi mr}\right)}$$\displaystyle T=\dfrac{2\pi r(2\pi mr)}{nh}$$\displaystyle T=\dfrac{4\pi^{2}mr^{2}}{nh}$From $\displaystyle r=\dfrac{n^{2}h^{2}}{4\pi^{2}mke^{2}}$$\displaystyle T=\dfrac{4\pi^{2}m}{nh}\left(\dfrac{n^{2}h^{2}}{4\pi^{2}mke^{2}}\right)^{2}$$\displaystyle T=\dfrac{n^{3}h^{3}}{4\pi^{2}mk^{2}e^{4}}$$\displaystyle \Rightarrow T\propto n^{3}$Alternatively ---$\displaystyle T=\dfrac{2\pi r}{v}$$\displaystyle \because r\propto n^{2}$and $\displaystyle v\propto\dfrac{1}{n}$$\displaystyle \therefore T\propto n^{3}$