Bohr magneton

Magnetic dipole moment of a revolving electron

In hydrogen- like atoms, an election revolves around the nucleus. It's motion is equivalent to a current loop which possesses a magnetic dipole moment = IA. Consider an electron revolving anticlockwise around a nuclues in an orbit of radius r with speed v and time period T. 

   

   Equivalent current, 

         I = charge/time = e/T = e/2πr/v = ev/2πr

        Area of the current loop, 

          A = πr × r

The orbital magnetic moment of the electron is

                Magnetic moment = IA = ev. πr × r/2πr

            Magnetic moment = evr/2        ...... (1) 

     

 According to right hand thumb rule, the direction of the magnetic dipole moment of the revolving electron will be perpendicular to the plane of it's orbit and in the downward direction, 

 Also, the angular momentum of the electron due to it's orbital motion is 

           l = mass of electron × vr     ..... (2) 

The direction of angular momentum is normal to the plane of the electron orbit and in the upward direction, 

 Dividing equation (1) by (2), we get 

       Magnetic moment/l = evr/2/mass of electron × vr = e/2 × mass of electron

The above ratio is a constant called gyromagmetic ratio. It's value is 8888888888 × 10 C/ kg.  So

    

      Magnetic moment = el/2 × mass of electron

   Vectorially,   

   

      Magnetic moment in vector form = - e × angular momentum in vector form /2 × mass of electron

 The negative sign shows that the direction of  angular momentum  is opposite to that of  magnetic moment. According to boards quantisatiom condition, the angular momentum of an electron in any permissible orbit is, 

   l = nh/2π,          where   n = 1,2,3,.....

   

  Magnetic moment = n( eh/4π × mass of electron) 

  This equation gives orbital magnetic moment of an electron revolving in nth orbit. 

 Bohr magneton. It is defined as the magnetic moment associated with an electron due to it's orbital motion in the first orbit of hydrogen atom. It is the minimum value of magnetic moment which can be obtained by putting n = 1 in the above equation. Thus bohr magneton is given by 

       = minimum value of magnetic moment = eh/4π × mass of electron = 9.27 /10 of power of 24   A(m of power of 2)