The ratio of acceleration of the electron in a singly ionized Helium atom $(He^+)$ to that of a Hydrogen atom $(H)$ (both in the ground state) is:

In a hydrogen atom,an electron of mass $9.1 \times 10^{-31} \ kg$ revolves about a proton in a circular orbit of radius $0.53 \ \mathring{A}$. The radial acceleration and angular velocity of the electron are respectively:

In $(i)$ a hydrogen atom and $(ii)$ a singly ionized helium atom,an electron makes a transition from the same excited state $n$ to the ground state. The ratio of the wavelengths of the emitted photons in the two cases will be:

The time period of revolution of an electron in the $n^{\text{th}}$ orbit in a hydrogen-like atom is given by $T = \frac{T_0 n^a}{Z^b}$,where $Z$ is the atomic number. Identify the correct values for $T_0$,$a$,and $b$.

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 ionization potential of a hydrogen-like atom is $122.4 \, V$. Find its atomic number $Z$.