# Electric Current & Drift Velocity | Relation

## Relation Between Electric Current and Drift Velocity-

Consider a conductor having-

• Length of conductor = L
• Area of cross-section = 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   …………….(Equation-01)

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 vd, then time taken by the electrons to cross the length of the conductor is given by- (Equation-02)

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- (Equation-03)

where-

• n = free electron density
• e = charge on an electron
• A = area of cross-section of the conductor
• Vd = drift velocity

## Relation Between Electric Current and Relaxation Time-

From Equation-03, the relation between electric current and drift velocity is- (Equation-03)

Also, we know the relation between drift velocity and relaxation time is- (Equation-04)

Using Equation-04 in Equation-03, we get- Thus, the relation between electric current and relaxation time is- (Equation-05)

where-

• n = free electron density
• e = charge on electron
• A = area of cross-section
• E = electric field intensity
• T = relaxation time
• m = mass of electron

## Deduction of Ohm’s Law-

From Equation-05, we have- (Equation-05)

If V is the potential difference applied across the two ends of the conductor, then we have- (Equation-06)

Using Equation-06 in Equation-05, 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, (Equation-07)

R is called as resistance of the conductor.

## Relation Between Resistivity and Relaxation Time-

From Equation-07, we have- (Equation-07)

Also we know, the resistance R of a conductor having length L, cross-sectional area A and resistivity ρ is given by- (Equation-08)

On comparing Equation-07 and Equation-08, we get- This is the relation between resistivity and relaxation time.

### Important Note

The resistivity of material of the conductor is independent of the dimensions of the conductor.

But it depends upon the following two parameters-

• Free electron density of the conductor
• Relaxation time