We all know the simplest version of Ohm's law:
where is the voltage drop across a resistor of resistance when a current flows through it. Let us generalize this law so that it is expressed in terms of and , rather than and . Consider a length of a conductor of uniform cross-sectional area with a current flowing down it. In general, we expect the electrical resistance of the conductor to be proportional to its length, and inversely proportional to its area (i.e., it is harder to push an electrical current down a long rather than a short wire, and it is easier to push a current down a wide rather than a narrow conducting channel.) Thus, we can write
The constant is called the resistivity, and is measured in units of ohm-meters. Ohm's law becomes
However, (supposing that the conductor is aligned along the -axis) and , so the above equation reduces to
There is nothing special about the -axis (in an isotropic conducting medium), so the previous formula immediately generalizes to
This is the vector form of Ohm's law.
Hope this helps.