The ohm’s law can be derived from the equation J= σE. Consider a segment of wire of length l and cross sectional area A as shown in figure

When a potential difference Vis applied across the wire, a net electric field is created in the wire which constitutes the current in the wire. For simplicity, we assume that the electric field is uniform in the entire length of the wire, then the potential difference (voltage V) can be written as

V=El

As we know, the magnitude of current density

eqn (1)

J=\sigma E=\sigma \frac{V}{l}

But J=\frac{I}{A} so we write the equation 1 as

By rearranging the above equation, we get

eqn (2)

The quantity \frac{l}{\sigma A} is called resistance of the conductor and it is denoted as R . Note that the resistance is directly proportional to the length of the conductor and inversely proportional to area of cross section.

Therefore, the macroscopic form of ohm’s law can be stated as

eqn (3)

V=IR 

From the above equation, the resistance is the ratio of potential difference across the given conductor to the current passing through the conductor.

eqn (4)

Materials for which the current versus voltage graph is a straight line through the origin, are said to obey Ohm’s law and their behaviour is said to be ohmic as shown in Figure. Materials or devices that do not follow Ohm’s law are said to be non-ohmic. These materials have more complex relationships between voltage and current. A plot of I versus V for a non-ohmic material is non-linear and they do not have a constant resistance.