Find an expression for the magnetic dipole moment and magnetic field induction at the center of Bohr's hypothetical hydrogen atom in the nth orbit of the electron in terms of universal constant.

Answer

B, C

Explanation

Magnetic moment: According to Bohr's hypothesis, angular momentum in `nth` orbit is `L=n(h/(2pi))`
Further, `M/L=q/(2n)` or `e/(2m)`
`:.` Magnetic moment,
`M=(e/(2m))(L)=(e/(2m))=(n h/(2pi))=((neh)/(4pim))`
Magnetic field induction
`(mv^2)/r=1/(4piepsilon_0)((e)(e))/r^2`..........i
From Bohr's hypothesis:
`mvr=(nh)/(2pi)`.........ii
Solving these two equations, we find
`v=e^2/((2epsilon_0nh)` and `r=(epsilon_0n^2h^2)/(pime^2)`
Now, magnetic field induction at centre
`B=(mu_0i)/(2r)`
Here, `i=qf=(e)(v/(2pir))`
`:. B=(mu_0e)/(4pi)(e^2/(2epsilon_0nh))(pi^2m^3e^4)/(epsilon_0^2n^4h^4))`
`=(mu_0pim^2e^7)/(8epsilon_0^3h^5n^5)`