Drift velocity of electron and resistance


1) Suppose a conducting metal has area of cross-section A.



2) Since this is a conductor, it has free electrons.


3) Now free electrons can move inside the metal, but they can not escape from the metal,


4) When free electrons move they collide metal ions every moment.


5) when an electric field is applied the electrons starts moving but as they collide with the metal ions, this collision with metal ions develops resistance.


6) This is the reason why resistance R is offered by a conductor in an electric field.  


7) free electrons are moving randomly inside the metal, so total average velocity of electrons is zero.


8) Say there are n numbers of electrons with velocity, u1,  u2,  u3,  u4, ………………………..   un  



Then




9) Suppose an electric field E  is applied.



10) The force acting on an electron is

= - e E



11) Then acceleration of the electron is





me is the mass of an electron


12) Let t1 is the time between a successive collisions of first electron. This time is called the relaxation time of an electron.



13) So the velocity of an electron due to this electric field is

v = u + at


for 1st electron



For 2nd  electron




For 3rd  electron



……………………………………

……………………………………

For nth  electron



 

14) therefore the average velocity of electrons is

 













Where



t is called the average relaxation time of electrons.

15) this velocity Vd  is called the drift velocity of electron.



Here -ve sign indicates that this velocity is opposite to the direction of electric field.

Therefore magnitude of drift velocity is