If $E$ and $L$ denote the magnitude of total energy and angular momentum of a revolving electron in the $n^{\text{th}}$ Bohr orbit,then:

Which of the following graphs in the figure shows the speed $(v)$ of an electron in a hydrogen atom as a function of the principal quantum number $(n)$?

$A$ 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 $v \propto n^{1/3}$
$(C)$ $E = \frac{3}{2} \left( \frac{n^2 h^2 F^2}{4 \pi^2 m} \right)^{1/3}$
$(D)$ $E = 2 \left( \frac{n^2 h^2 F^2}{4 \pi^2 m} \right)^{1/3}$

Let $r, v, E$ be the radius of orbit, speed of electron, and total energy of electron respectively in a $H$-atom. Which of the following quantities, according to Bohr theory, is proportional to the quantum number $n$?

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

The maximum wavelength that a sample of hydrogen atoms can absorb is