An electron is in a circular orbit about the nucleus of an atom. Find the ratio between the orbital magnetic dipole moment `vec mu` and the angular momentum `vec L` of the electron about the center of its orbit.

Explanation

Let `v` be the speed of the electron and r the radius of its orbit.
The orbating electron creates an electric current opposite to the velocity. The current is the charge that passes any given point along the circumferential path during one second. Thus, current is charge e times the frequency f of the orbital motion.
`I=ef=e/T=(ev)/(2pir)`
The magnitude of magnetic moment,
`mu=IA=(ev)/(2pir)pir^2=(evr)/2`
The direction of the magnetic dipole moment is perpendicular to the loop from right hand thumb rule.
The orbital angular momentum,`vecL_("orbit")=vec r xx vec p`
`|vecL_("orbit")|= rp sin90^@=mvr`
The direction of the orbital angular momentum is determined from the vector product right hand rule. The directions of angular momentum and angular momentum and magnetic dipole moment are opposite to each other.
`vecmu=-e/(2m)vecL`
.