Question

1)suppose we start with an electron with zero initial velocity. Let ϝ be

the typical time it would take for our electron to hit an atom. We can use this as

an average time between collisions as the electron makes its way through the

material. What is the typical velocity of the electron when it hits an atom?

2) It turns out that the velocity you just calculated is not a bad estimate of the drift

velocity, vd. Now, as we saw, there is another way to write vd, namely

J = (❝ne)vd for electrons (remember J = i / A for uniform current density), where

n is the number of charge carriers (electrons) per unit volume. Solve this for |vd|

and equate with what you got in (1).

3)In the above, you should have a J on one side and an E on the other. Now J and E
are in another important equation from today. Given that, can you find an
expression for the resistivity of the material, ??

4)Also, we just saw in lecture that R = (L/A)?, where L is the length of the wire and
A is its cross-sectional area. So, what

Answer 1

1) E = electric filed   , e = charge on electron

     a = acceleration = eE/m

V = velocity = a*t = eE*t/m

2) J = neVd

     Vd = J/(n*e)

given   Vd = V

         J/(n*e) =   eE*t/m

           

           J   = ne( eEt/m)

      

        let   et/m = mobility = mu

             hence    J = ne * mu * E

       

3)    as we know : J = sigma * E

                       where sigma = concutivity

                                      then sigma = ne * mu

                              sigma   = n*e*et/m = ne^2*t/m

                        

                 p = resistivity = 1/sigma     = m/[ ne^2*t ]

4)        R =( L/A)*p

               =(L/A)* [m/(ne^2t)]

   5) sigma (conductivity) , indipendent of applied electric filed .

   Hence R is independent from E lectric field