1.2) Electron As A Particle: The Hall Effect

                            The Hydrodynamic Model of Electron Flow

The motion of equation of an electron is given by:

                                                  m(dv/dt) = eE             ……(1.13)

If we assume that the electron moves in a viscous medium then the ‘momentum destroying’             term ηv accounts for the resistance to the motion of the electron. Hence, the equation of motion to describe the above model can be given by:

                                            m((dv/dt) + ηv)  = eE             ……(1.14)

When viscosity dominates, the dv/dt term becomes negligible, and the equation o motion is then given by:

                                                      mηv = eE                     ….(1.15) 

Clearly, we can see that η = 1/τ equation (1.15) agrees with equation (1.6). Hence the two models can be treadted as equivalent and, in any given case, use whichever is convenient.

                                           The Hall Effect

In the given figure the current flows in the negative x-direction in the rectangular bar, thus the motion of electrons is in the positive x-direction.  The magnetic field is in the positive z-direction. Thus the force on an electron  is given by:

                                                   F_m = e(v x B)               ……(1.16)

Hall Effect

The (v x B) is in the negative y-direction, but e, the charge on an electron is negative, hence the direction of F_m is in the positive y-direction. Thus the electrons are deflected in the positive y-direction. This results in a negatively charged and a positively charged end of the bar, hence creating an electric field E in the positive y-direction. The force  F_E acts in the negative y-direction. At one stage an equilibrium is attained and the magnitude of F_E and F_m become equal. At this point the magnitude of E will be given by:

                                                               E = vB           ……(1.17)

But from (1.7) we get :

                                                             E = (JB)/(N_ee)    

                                                             E = R_HJB               ……(1.18)   

In the experiment, E, J and B are known. Thus N_e (electron density) can be calculated.

Once the polarity of the applied voltage and magnetic field are fixed, the electric field is well defined. So for a particular set of applied voltage and magnetic field, the measured transverse voltage should  have the same polarity for any conductor. According to experimental facts for some conductors and semiconductors the measured transverse voltage is in the opposite direction. This means that in such materials electricity is carried by positively charged particles called holes. They are not actually particles but vacancies, but for the time being we shall consider them as particles.

Thus there are now two species of charge carriers bouncing around in our model, which can be considered as a mixture of two gases. The net charge is kept zero and the new model is ready.