Define the current density of a metallic conductor. Deduce the relation between current density and conductivity of the conductor when an electric field is applied to it.

Solution

1. Definition of Current Density

Current density (J) is defined as the amount of electric current flowing per unit cross-sectional area of a conductor.

J = I / A

2. Physical Meaning

Current density is a vector quantity. Its direction is the same as the direction of flow of positive charge (conventional current).

3. Derivation of Relation between Current Density and Conductivity

Let:

n = number density of free electrons

e = charge of electron

m = mass of electron

τ = relaxation time

E = electric field

Drift velocity:

v_d = (eEτ) / m

Current density:

J = n e v_d

Substitute v_d:

J = n e (eEτ / m)

J = (n e² τ / m) E

From Ohm’s law in microscopic form:

J = σE

Comparing both expressions:

σ = (n e² τ) / m

Hence, relation becomes:

J = σE

Final Result (To Remember)

J = σE

σ = (n e² τ) / m