The wavelength of light emitted when an electron transitions from the second orbit to the first orbit in a hydrogen atom is:

For a hydrogen atom,when an electron jumps from $n = 2$ to $n = 1$,the wavelength of the radiation emitted is found to be $\lambda_0$. For which transition of an electron in a $He^+$ ion will the wavelength of the radiation emitted be equal to $\lambda_0$?

Hydrogen $(H)$,deuterium $(D)$,singly ionized helium $(He^+)$ and doubly ionized lithium $(Li^{++})$ all have one electron around the nucleus. Consider the $n = 2$ to $n = 1$ transition. The wavelengths of emitted radiations are $\lambda_1, \lambda_2, \lambda_3$ and $\lambda_4$ respectively.

According to Bohr's theory,the moment of momentum of an electron revolving in the $4^{\text{th}}$ orbit of a hydrogen atom is:

Classically,an electron can be in any orbit around the nucleus of an atom. Then what determines the typical atomic size? Why is an atom not,say,thousand times bigger than its typical size? The question had greatly puzzled Bohr before he arrived at his famous model of the atom that you have learnt in the text. To simulate what he might well have done before his discovery,let us play as follows with the basic constants of nature and see if we can get a quantity with the dimensions of length that is roughly equal to the known size of an atom $\left(\sim 10^{-10} \;m \right)$.
$(a)$ Construct a quantity with the dimensions of length from the fundamental constants $e, m_{e},$ and $c$. Determine its numerical value.
$(b)$ You will find that the length obtained in $(a)$ is many orders of magnitude smaller than the atomic dimensions. Further,it involves $c$. But energies of atoms are mostly in non-relativistic domain where $c$ is not expected to play any role. This is what may have suggested Bohr to discard $c$ and look for 'something else' to get the right atomic size. Now,the Planck's constant $h$ had already made its appearance elsewhere. Bohr's great insight lay in recognising that $h, m_{e},$ and $e$ will yield the right atomic size. Construct a quantity with the dimension of length from $h, m_{e},$ and $e$ and confirm that its numerical value has indeed the correct order of magnitude.

An electron is revolving in a circular orbit of radius $r$ in a hydrogen atom. The angular momentum of the electron is $L$. The relation between the magnetic dipole moment $(m)$ associated with it,the gyromagnetic ratio $(R)$,and $L$ is: