Current Density and Drift Velocity

The current flowing through a unit cross-sectional area of a conductor is called current density (J).

Mathematically, If 'dI' current passes through a cross-sectional area ‘dt’ of a conductor then current density may be written as

Current Density = J = dI/da ➔ 1

If the current is measured in ‘Ampere and area in ‘meter’ then

J = Amphere/(meter)² = A/m²

J = Am⁻²

The unit of current density is Am⁻².

From Equation 1, we can write

J = dI/da ⇒ dI = Jda ➔ 2

Integrating equation 2

∫ dI = ∫ J.da

I = J ∫ da

I = JA

J = I/A

Current density is a vector quantity and its direction is the same as that of electric field Ē.
When a conductor is connected to a battery, the free electrons move from low potential to high potential i.e from the negative terminal to the positive terminal.

During motion, the force electrons collide with the ionic cores of the conductor. The K.E of the accelerating electrons is converted into vibrational energy of the lattice the constant average velocity acquired by free electrons against the electric field is called Drift Velocity V𝒹.

It is of the order 10⁻³ms⁻¹. In order to calculate the magnitude of drift velocity, let us take a conductor of length 'L' and area of cross-sectional 'A' then the volume of the conductor is equal to AXL.

The volume of the conductor = AXL ➜ 1

If the unit volume of the conductor carries n charge careers then
Total no of charge careers in the conductor = n (AXL) ➟ 2

And if each charge career carries a charge 'e' then
Total charge on the conductor = e [n(AL)]

q = e (nAL) ➜ 3

When this conductor is connected to a voltage source, the charge careers will start to flow from the negative terminal to the positive terminal with an orbit velocity V𝒹.

If 't' is the time taken by the charge careers to pairs through the whole length (1) of the conductor then we can white.

S = v x t

L = V𝒹 x t

t = L/V𝒹 ➜ 4

Due to the motion of these charge careers, a current 'I' will be established whose magnitude is given as

I =q/t ➜ 5

Putting the value of q and t from equations 3 and 4 into above equation 5 we have

I = e (nAL) / L/V𝒹

I = e (nA)V𝒹

V𝒹 = I/ (nA)e

V𝒹 = I/A x 1/ne

V𝒹 = J x 1/ne ➜ 6

The above equation correlates the drift velocity with current density. Since the moving charge careers in a conductor are electrons so the above equation can be rewritten as

V𝒹 = J / (-e)n

J = en (-V𝒹)

Thus the direction of drift velocity and current density are opposite to each other.

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