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The noramal Zeeman effect can be explained by Classical electron theory as well as Quantum theory without taking electron spin.

Now, let us explain Noraml Zeeman effect with the help of mathematical expression. 

Let us explain it by Classical Theory or Lorentz Classical electron Theory.
Consider an electron moving in a circular orbit of radius “r” with a velocity “v” ,the centripital force acting on electron is given by :

When an external magnetic field B is applied perpendicular to the direction of motion of the electron, it will experience a Lorentz force i.e.

Now, we have:

Also, we have,



The quantity eB/4πm is known as Normal Zeeman Sepration.

Zeeman  Shift :

The quantity dυ  = ±eB/4πm is called Zeeman Shift or Zeeman Sepration.

Thus,  when light is seen perpendicular to the magnetic field, TV hen three components ” υ  – dυ  ” , ” υ ” , ”  υ + d υ ” gives a normal triplet.
In terms of an atom consists of  orbital angular momentum L and magnetic moment μl
Then we have, 

     ,where e and m are charge & mass of electron.
The μl and L are oppositely directed when the atom is placed in magnetic field  B which is acting along z-axis, when L precesses around the magnetic field ( Larmor Precession)  with quantised components. 

When the atom is placed in external magnetic field then quantised components are given by :

By Larmor’s theorem,  the angular velocity of Precession is given by :

The interaction energy of such precession is equal to product of angular velocity and projection of L along that field, i.e.

Now,  ml takes (2l + 1 ) discrete values. It is each energy level for a given value of l,  is shifted into ( 2l + 1) energy levels called Zeeman level. 

Let us consider, a line arising from l = 1 to l = 0 level in the magnetic field. The l = 1 level is splitted into (2l + 1 ) =  2 × 1 + 1 = 3 levels corresponding to ml = -1, 0, +1, while the level l = 0 

=> 2(0) +1 = 1 level, i.e. this level is unstable.

 original line of frequency υ0 is shifted to :
where, υ0  is frequency of unshifted components & υ1 and υ2  are frequency of shifted components. 

The change in frequency in noramal Zeeman effect is given by :

Thus,  measuring change of wavelength  λ we can find the value of e/m.