In a hydrogen-like atom with atomic number $Z$,an electron is in an excited state with principal quantum number $2n$. The maximum energy of a photon that can be emitted from this state is $204 \ eV$. If the electron transitions from the $2n$ orbit to the $n$ orbit,a photon with energy $40.8 \ eV$ is emitted. The value of $n$ is:

If the energy in the first excited state in a hydrogen atom is $23.8 \, eV$,then the potential energy of a hydrogen atom in the ground state can be assumed to be ....... $eV$.

$A$ particle of mass $m$ moves in circular orbits with potential energy $V(r) = Fr$,where $F$ is a positive constant and $r$ is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle's orbit is denoted by $R$ and its speed and energy are denoted by $v$ and $E$,respectively,then for the $n^{\text{th}}$ orbit (here $h$ is the Planck's constant)-
$(A)$ $R \propto n^{2/3}$ and $v \propto n^{1/3}$
$(B)$ $R \propto n^{2/3}$ and $v \propto n^{1/3}$
$(C)$ $E = \frac{3}{2} \left( \frac{n^2 h^2 F^2}{4 \pi^2 m} \right)^{1/3}$
$(D)$ $E = 2 \left( \frac{n^2 h^2 F^2}{4 \pi^2 m} \right)^{1/3}$

The ground state energy of a hydrogen atom is $-13.6 \ eV$. In this state,the potential energy is . . . . . . $eV$.

The radius of the orbit of an electron in a hydrogen atom is $0.5 \ \mathring{A}$. The speed of the electron is $2 \times 10^6 \ m/s$. The current in the loop due to the motion of the electron is ............. $mA$.

The ratio of radii of $3^{\text{rd}}$ and $6^{\text{th}}$ Bohr's orbit in a hydrogen atom is