In the Bohr model of the hydrogen atom,let $R, v$,and $E$ represent the radius of the orbit,the speed of the electron,and the total energy of the electron,respectively. Which of the following quantities is proportional to the quantum number $n$?

The ratio of the kinetic energy to the total energy of an electron in a Bohr orbit is

In the Bohr model of a hydrogen-like atom,the force between the nucleus and the electron is modified as $F = \frac{e^2}{4\pi \varepsilon_0} \left( \frac{1}{r^2} + \frac{\beta}{r^3} \right)$,where $\beta$ is a constant. For this atom,the radius of the $n^{th}$ orbit in terms of the Bohr radius $\left( a_0 = \frac{\varepsilon_0 h^2}{m \pi e^2} \right)$ is:

Find the ratio of the area of the orbit of the first excited state to the ground state in a hydrogen atom.

If an electron in a hydrogen atom jumps from the third orbit to the second orbit,the frequency of the emitted radiation is given by (where $c$ is the speed of light and $R$ is the Rydberg constant):

The kinetic energy of an electron revolving around a nucleus will be