Magnetic field at the centre of the hydrogen atom due to the motion of an electron in the $n^{\text{th}}$ orbit is proportional to:

$P$ is the probability of finding the $1s$ electron of a hydrogen atom in a spherical shell of infinitesimal thickness, $dr$, at a distance $r$ from the nucleus. The volume of this shell is $4\pi r^2 dr$. The qualitative sketch of the dependence of $P$ on $r$ is:

An electron revolving in the $n^{\text{th}}$ Bohr orbit has a magnetic moment $\mu_n$. If $\mu_n \propto n^x$,the value of $x$ is:

The de-Broglie wavelength of an electron in the $3^{rd}$ orbit of a $He^{+1}$ ion is approximately (in $\mathring{A}$):

The potential energy of a proton and an electron in a hydrogen atom is given by $V = V_0 \ln \left( \frac{r}{r_0} \right)$,where $r_0$ is a constant. Assuming the system follows the Bohr model,find the relationship between the radius $r_n$ and the principal quantum number $n$.

In any Bohr orbit of the hydrogen atom,the ratio of kinetic energy to potential energy of the electron is