Find the following expression for magnetic dipole moment of an electron moving in a circular path.
`mu_(e )=(e )/(2m_(e ))l`
What is gyromagnetic ratio?

Explanation

An electron being a charged particle, constitutes a current, while moving in its circular orbit around the nucleus (`:.`moving charge constitutes a current as well as magnetic field). If T is the time period of revolution, then current constituted by electron is `I= (e )/(T ) " " ...(i)`
where, e = charge of electron.
If r is the orbital radius of electron and its orbital speed is v, then
`T=(2pi r)/(v)`
`rArr I=(e )/((2pi r)/(v)) =(e v)/(2pi r) " " ` [from Eq. (i)]
Magnetic moment of revolving electron, M=IA
`M=(ev)/(2pi r) pi r^(2) =(ev r)/(2)`
The direction of this magnetic moment is into the plane of the paper.
`M=(e )/(2m_(e )) (m_(e )vr) =(e )/(2m_(e )) l`
or `M=(- e l)/(2m_(e ))`
where, `l=m_(e )vr` is angular momentum of the electron.
`(M)/(l) =(e )/(2m_(e ))` is a constant, called gyromagnetic ratio, its value is `8.8xx 10^(10)C//kg` for an electron.