com www.GOALias.blogspot.com

www.GOALias.blogspot.comPhysicstime more than τ and some less than τ. In other words, the time t iinEq. (3.16) will be less than τ for some and more than τ for others as we gothrough the values of i = 1, 2 ..... N. The average value of t ithen is τ(known as relaxation time). Thus, averaging Eq. (3.16) over theN-electrons at any given time t gives us for the average velocity v dev ≡ V = v − Em( ) ( ) ( t )d i average i average i average0– e E e E= τ =− τ(3.17)m mThis last result is surprising. It tells us that theelectrons move with an average velocity which isindependent of time, although electrons areaccelerated. This is the phenomenon of drift and thevelocity v din Eq. (3.17) is called the drift velocity.Because of the drift, there will be net transport ofcharges across any area perpendicular to E. Considera planar area A, located inside the conductor such thatFIGURE 3.4 Current in a metallic the normal to the area is parallel to Econductor. The magnitude of current (Fig. 3.4). Then because of the drift, in an infinitesimaldensity in a metal is the magnitude of amount of time ∆t, all electrons to the left of the area atcharge contained in a cylinder of unit distances upto |v d|∆t would have crossed the area. Ifarea and length v d.n is the number of free electrons per unit volume inthe metal, then there are n ∆t |v d|A such electrons.Since each electron carries a charge –e, the total charge transported acrossthis area A to the right in time ∆t is –ne A|v d|∆t. E is directed towards theleft and hence the total charge transported along E across the area isnegative of this. The amount of charge crossing the area A in time ∆t is bydefinition [Eq. (3.2)] I ∆t, where I is the magnitude of the current. Hence,I ∆ t = + ne A v ∆t(3.18)dSubstituting the value of |v d| from Eq. (3.17)2e AI ∆ t = n tm τ ∆ E (3.19)By definition I is related to the magnitude |j| of the current density byI = |j|A (3.20)Hence, from Eqs.(3.19) and (3.20),2j = nem τ E (3.21)The vector j is parallel to E and hence we can write Eq. (3.21) in thevector form982j=nem τE(3.22)Comparison with Eq. (3.13) shows that Eq. (3.22) is exactly the Ohm’slaw, if we identify the conductivity σ as