Physics - Modern Physics Question with Solution | TestHub

According to Bohr's model, the radius of the $n^{\text{th}}$ orbit of a hydrogen atom is given by:

$r=\frac{{\epsilon }_0{n}^2{h}^2}{\pi m{e}^2}$

The velocity of the electron in the $n^{\text{th}}$ orbit is given by:

$v=\frac{e^2}{2{\epsilon }_{0}nh}$

The orbital period $T$ is the time taken for one complete revolution, which is:

$T=\frac{2\pi r}{v}$

Substituting the expressions for $r$ and $v$ into the equation for $T$:

$T=\frac{2\pi \left(\frac{{\epsilon }_0{n}^2{h}^2}{\pi m{e}^2}\right)}{\frac{e^2}{2{\epsilon }_{0}nh}}$

$T=\frac{2{\epsilon }_0{n}^2{h}^2}{m{e}^2}\times \frac{2{\epsilon }_{0}nh}{e^2}$

$T=\frac{4{\epsilon }_0^2{n}^3{h}^3}{m{e}^4}$

Answer: $\frac{4{\epsilon }_0^2{\mathrm{n}}^3{\mathrm{h}}^3}{m{\mathrm{e}}^4}$