Expression for Electrical Conductivity

EXPRESSION FOR ELECTRICAL CONDUCTIVITY

We know in the absence of external electric field, the motion of electrons in a metal moves randomly in all directions. When an electric field (i.e.,) potential difference is maintained between the two ends of a metallic rod, as shown in Fig. 1.4 the electrons will move towards the positive field direction and produces the current in the metallic rod.

If 'n' is the free electron density and ‘e' is the charge of electron then the current density (i.e.,) the current flowing through unit area is given by

J = n v_d (-e) ….(1)

The - ve sign implies that the charge of the electron is negative and it also indicates that the conventional direction of current is in the opposite direction to the electron movement.

Due to the electric field applied, the electron gains the acceleration 'a'

(or) v_d =aτ ….(2)

If E is the electric field intensity and 'm' is the mass of the electron, Then,

The force experienced by the electron is F =- eE ...(3)

From Newton's Second law of motion,

The force on the electron F = ma ...(4)

Equating equation (3) and equation (4) we have

- eE= ma 

...(5)

Substituting equation (5) in equation (2) we have

….(6)

Substituting equation (6) in equation (1) we have

….(7)

Here the number of electrons flowing per second through unit area (i.e.,) the current density, depends on the field applied. Thus if the field (E) applied is more, current density (J) will also be more

We can write J α E (or) J = σ E ….(8)

If A = 1; E = 1 then

σ = 1.

Definition: Coefficient of electrical conductivity (o) which is defined as the quantity of electricity flowing per unit area per unit time maintained at unit potential gradient. Unit: Ω ^-1m^-1

Comparing equation (7) and equation (8) we can write

….(9)