Imagine removing one electron from $He^4$ and $He^3$. Their energy levels,as worked out on the basis of the Bohr model,will be very close. Explain why?

Electrons in a certain energy level $n=n_{1}$ can emit $3$ spectral lines. When they are in another energy level $n=n_{2}$,they can emit $6$ spectral lines. The orbital speed of the electrons in these orbits are in the ratio:

Monochromatic radiation is incident on a hydrogen $(H)$ sample which is in the ground state. If the hydrogen atoms emit radiation of $10$ different wavelengths after absorbing the incident radiation, then the wavelength of the incident radiation is (Let $hc = 1242 \text{ eV-nm}$) (in $\text{ nm}$)

The magnetic moment of an orbital electron in the first orbit of an $\text{H}$-atom is $\mu_{B}$. What is the magnetic moment of that electron in the second orbit?

The radius of the first orbit of hydrogen is $r_{H}$,and the energy in the ground state is $-13.6 \text{ eV}$. Considering a $\mu^{-}$-particle with a mass $207 m_e$ revolving around a proton as in a hydrogen atom,the energy and radius of the proton and $\mu^{-}$-combination respectively in the first orbit are (assume the nucleus to be stationary):

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.