Question 12

Using the concept of free electrons in a conductor, derive the expression for the conductivity of a wire in terms of number density and relaxation time. Hence obtain the relation between current density and the applied electric field E.

Answer

Consider a conductor of length L and cross sectional area A , when potental difference v is applied across its end the current produced is i. If n is the number of electrons per unit volume in conductor and V_{d} is the drift velocity then,

I = -nAeV_{d } Equaltion 1.

Where e is charge on electron , n is no of density

Electric field produced at each point, E = V/L Equation 2

if t is relaxation time and e is the electric field strength then drift velocity

Vd = -etE / m Equation 3

Substituting value in equation 1:

I = - neA(-etE / m )

I = (n e2t / m ) AE Equation 4

As e = V/L then,

I = n e2t A / m x V/L

Or

V/I = m/n e2t x L/A

From other law R = V/I = m/n e2t x L/A Equation 5

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