Imagine an atom made up of a proton and a hypothetical particle of double the mass of the electron but having the same charge as the electron. Apply the Bohr model to this atom. The longest wavelength photon that will be emitted has wavelength $\lambda$ (given in terms of the Rydberg constant $R$ for the hydrogen atom) equal to:

The following statements are given about the hydrogen atom:
$A$. The wavelengths of the spectral lines of the Lyman series are greater than the wavelength of the second spectral line of the Balmer series.
$B$. The orbits correspond to circular standing waves in which the circumference of the orbit equals a whole number of wavelengths.

$A$ hydrogen atom is in its $n^{\text{th}}$ energy state. If the de-Broglie wavelength of the electron is $\lambda$,then:

The ratio of areas within the electron orbits for the first excited state to the ground state for a hydrogen atom is (in $ : 1$)

From a carbon nanotube of $1 \,\mu m$ length and $1 \,nm$ radius,$10$ electrons have been removed. Assume the resulting positive charge to be distributed uniformly over the surface of the tube. The energy of an electron moving in a stable circular orbit around the axis along the length of the tube is calculated by applying the Bohr model. Accordingly,the frequency of radiation required to excite an electron from its ground state to the next level is in the range of (charge of the electron,$e = 1.60 \times 10^{-19} \,C$; mass of the electron,$m_e = 9.11 \times 10^{-31} \,kg$; Planck's constant,$h = 6.63 \times 10^{-34} \,Js$; Permittivity of free space,$\varepsilon_0 = 8.85 \times 10^{-12} \,F/m$)

The frequency of light emitted,when the electron makes a transition from the level of principal quantum number $n=2$ to the level with $n=1$ is (Take,the ionization energy of hydrogen to be $13.6 \ eV$ and $h \simeq 4 \times 10^{-15} \ eV \cdot s$)