Relation Between Electric Current and Drift Velocity
Consider a conductor having
 Length of conductor = L
 Area of crosssection = A
 Number of free electrons per unit volume = n
Then, total number of free electrons inside the conductor,
N = Free electron density x Volume of the conductor
N = n x (A x L)
N = nAL
Total charge on free electrons inside the conductor,
Q = Total number of free electrons x Charge on one electron
Q = nAL x e
Q = nALe …………….(Equation01)
Now, when a battery is connected across the two ends of the conductor, an electric field is set up inside the conductor across its two ends.
If the Drift Velocity of electrons is v_{d}, then time taken by the electrons to cross the length of the conductor is given by
(Equation02)
We know, Electric Current is defined as the rate of flow of charge through the conductor i.e.
Substituting equations (1) and (2) here, we get
Thus, the relation between electric current and drift velocity is
(Equation03)
where
 n = free electron density
 e = charge on an electron
 A = area of crosssection of the conductor
 V_{d} = drift velocity
Relation Between Electric Current and Relaxation Time
From Equation03, the relation between electric current and drift velocity is
(Equation03)
Also, we know the relation between drift velocity and relaxation time is
(Equation04)
Using Equation04 in Equation03, we get
Thus, the relation between electric current and relaxation time is
(Equation05)
where
 n = free electron density
 e = charge on electron
 A = area of crosssection
 E = electric field intensity
 T = relaxation time
 m = mass of electron
Deduction of Ohm’s Law
From Equation05, we have
(Equation05)
If V is the potential difference applied across the two ends of the conductor, then we have
(Equation06)
Using Equation06 in Equation05, we get
At a fixed temperature, the quantities m, L, n, e, A and T, all have constant values for a given conductor. Therefore,
This proves Ohm’s law for conductor.
Here,
(Equation07)
R is called as resistance of the conductor.
Relation Between Resistivity and Relaxation Time
From Equation07, we have
(Equation07)
Also we know, the resistance R of a conductor having length L, crosssectional area A and resistivity ρ is given by
(Equation08)
On comparing Equation07 and Equation08, we get
This is the relation between resistivity and relaxation time.
Important NoteThe resistivity of material of the conductor is independent of the dimensions of the conductor. But it depends upon the following two parameters

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Resistance & Conductance of Conductor
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