Magnetic Dipole Moment of Revolving Electron Gyromagnetic Ratio

Description:

When an electron moves in a circular path there is a generation of electric current, this electric current is given by -

i = -e/T

The negative sign indicates that the direction of current is opposite to the motion of electron i.e. if the electron is revolving in anti-clockwise direction the current will have a clockwise direction.

Time period of revolution -

T = 2πr/V

Frequency of revolution -

υ = V/2πr

Angular velocity of revolution -

ω = V/r

Current can be expressed in various forms such as,

i = eV/2πr

i = ev

i = eω/2π

As revolving electron is generating current there will be a generation of magnetic field and it acts similar to magnetic field generated by current carrying circular coil.

Magnetic dipole moment of a current carrying circular coil is -

μ = NIA

For a single cycle N=1,

∴ μ = IA

Multiply and divide by m (m = mass of electron)

As we know angular momentum is 𝐿 = 𝑚𝑉𝑟

Gyromagnetic ratio

Ratio between magnetic dipole moment μ and Angular Momentum L is a constant known as gyromagnetic ratio and it is given by -

μ/L = e/2m

Observation

Whenever angular momentum of an electron increases its magnetic dipole moment also increases.

μ/L = 8.8 × 10^-10 C Kg^-1