Question

discuss magnetic dipole moment of an electron revolving in an orbit in hydrogen atom and derive the expression for the ratio of the magnetic dipole moment and the angular momentum as gyro magnetic ratio for the revolving electron​

Answer 1

To do:

Discuss magnetic dipole moment of an electron revolving in an orbit in hydrogen atom and derive the expression for GYRO-MAGNETIC ratio.

Calculation:

Magnetic dipole moment of an electron revolving in an orbit in hydrogen atom can be calculated by multiplying the current produced by the electron with the area of the circular orbit.

$\therefore \: M = i \times area$

$\implies\: M = \dfrac{q}{t} \times area$

$\implies\: M = \dfrac{e}{( \frac{2\pi r}{v} )} \times \pi {r}^{2}$

$\boxed{ \implies\: M = \dfrac{evr}{2}}$

Now, Gyromagnetic Ratio can be calculated as the ratio of the magnetic dipole moment and the angular momentum of the electron :

$\therefore \: GR = \dfrac{M}{L}$

$\implies\: GR = \dfrac{( \frac{evr}{2} )}{mvr}$

$\boxed{ \implies\: GR = \dfrac{e}{2m}}$

Hope It Helps.