Langevin’s Theory of Diamagnetism

Consider an electron (mass = m, charge = e) rotating about the nucleus (charge = Ze) in a circular orbit of radius r. Let ω₀ be the angular velocity of the electron. Then

F₀ = mω₀²r = Ze² /4π ε₀r²)
Or        ω₀ = √Ze² /4π ε₀mr³                         … (1)

The magnetic moment of the electron is
m = current x area = e ω₀/2π x πr² = e/2 ω₀r²                         … (2)

Let a magnetic field of induction B be now applied. B is normal to and in the page.

An additional force FL called Lorentz force acts on the electrons.
            FL = - e (vXB) = - eBrω
The condition of stable motion is now given by
            mrω2 = Ze2 /4πε₀r² - eBrω                    … (3)

or ω² + eB/mω - Ze² /4πε₀mr³) = 0
Solving the quadratic equation in ω,

ω= ((-eB)/m± √((eB/m)^2+4(Ze^2/(4πϵ₀ mr^3)/2

                    = ± √(ω₀^2+(eB/2m)^2 )-  eB/2m           

Or                 ω = + ω0 – eB/2m                (eB/2 << ω0)    … (4)

Thus the angular frequency is no different from ω0. Thus result of establishing a field of flux density B is to set up a processional motion of the electronic orbits with angular velocity – (e/2m)B. This is called Larmor theorem. Then