Question

Give reasons/short answer.

Prove that the current density of a metallic conductor is directly proportional to the drift speed of electrons.

Answer 1

1. Consider a part of conducting wire with its free electrons having the drift speed v_d in the direction opposite to the electric field `vec"E"`.
2. All the electrons move with the same drift speed v_d and the current I is the same throughout the cross-section (A) of the wire.
3. Let L be the length of the wire and n be the number of free electrons per unit volume of the wire. Then the total number of free electrons in the length L of the conducting wire is nAL.
4. The total charge in the length L is,
   q =  nALe......(1)
   where e is the charge of electron.
5. Equation (1) is the total charge that moves through any cross-section of the wire in a certain time interval t.
   ∴ t = `"L"/"v"_"d"` .......(2)
6. Current is given by,
   I = `"q"/"t"="nALe"/("L"//"v"_"d")` .....[From equations (1) and (2)]
   = n Av_de
   Hence, 
   v_d = `1/"n Ae"`
   = `"J"/"ne"` .......`(∵ "J"= 1/"A")`
   Hence for constant ‘ne’, current density of a metallic conductor is directly proportional to the drift speed of electrons, J α v_d.