Question

Find an expression for the magnetic dipole moment and magnetic field induction at the centre of a Bohr's hypothetical hydrogen atom in the $n^{th}$ orbit of the electron in terms of universal constants.

Answer 1

Magnetic moment: According to Bohr's hypothesis, angular momentum in nth orbit is:
$L=n(\frac{h}{2\pi })$

Further, $\frac{M}{L}=\frac{q}{2m}or\frac{e}{2m}$

$\therefore$ Magnetic moment, $M=(\frac{e}{2m}\left)\right(L)=(\frac{e}{2m}\left)\right(n\frac{h}{2\pi })=(\frac{neh}{4\pi m})$

Magnetic field induction :
$\frac{m{v}^2}{r}=\frac{1}{4\pi {\epsilon }_o}\frac{\left(e\right)\left(e\right)}{r^2}$ ........(i)
From bohr's hypothesis :
$mvr=\frac{nh}{2\pi }$ ......(ii)
Solving these two equations, we find,
$v=\frac{e^2}{2{\epsilon }_{o}nh}$ and $r=\frac{{\epsilon }_o{n}^2{h}^2}{\pi m{e}^2}$
Now magnetic field induction at centre
$B=\frac{{\mu }_{o}i}{2r}$
Here, $i=qf=\left(e\right)(\frac{v}{2\pi r}$
$\therefore B=(\frac{{\mu }_o}{2r}\left)\right(\frac{ev}{2\pi r})=\frac{{\mu }_{o}e}{4\pi }(\frac{v}{r^2})$
$=(\frac{{\mu }_{o}e}{4\pi }\left)\right(\frac{e^2}{2\epsilon nh}\left)\right(\frac{{\pi }^2{m}^2{e}^4}{{\epsilon }^2{n}^4{h}^4})$
$=\frac{{\mu }_o\pi m^2{e}^7}{8{\epsilon }^3{h}^5{n}^5}$