OHM Law and Electrical Conductivity

It is found experimentally that in a metallic conductor at constant temperature, the current density J linearly proportional to the electric field E. Thus
            J = σE

The equation J = σE is general vector statement of Ohm’s law. Here σ is a constant called the conductivity of the conductor. The unit of conductivity is mho/meter. The reciprocal of the conductor. The unit of conductivity is mho/meter. The reciprocal of the conductivity is called the resistivity p. In terms of p Eq. (1) can be written as
            E = pJ

The electric field along the rod is in the direction MN and its value is E = V/L everywhere. Here V is the total potential drop from M to N. Thus
J = v (V/L) The total current = I = JA = σ VA/L.

Here A is the cross-sectional area of the rod. This leads to

                V/i = L/ σA = pL/A

The ratio V/I is called the resistance R of the rod. R = pL/A.
Eq. (3) becomes V = iR

This is known as Ohm’s law.

Experiments show that generally p and therefore R increase with temperature according to
        R1 = R0 (1 + at)

where ά is called the temperature coefficient of resistivity, and t is the temperature in degrees centigrade.

The power dissipated in a resistance R carrying a current i is
        P = i² R = iV = V² / R

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