What mathematically describes Ohm's Law?

Answer 1

The original Ohm's law describes the relation between electrical potential (V) and current (I) at the extremes of a wire like element, where potential is applied on the wire extremes and current flows along the wire.

The law states that the current and the potential are proportional if the wire material is homogeneous and the proportionality constant is called the wire resistance R.

Thus it can be written as

V = R I

several generalizations of this law have been introduced to take into account more complex situations.

For example, if we consider the wire like a cylinder with a section area equal to A and a length L, we can introduce the resistivity of the wire (r) as the resistance of a wire having unitary section area and unitary length, so that the Ohm's law can be also written as

V = L r I /A

This form of the Ohm's law shows as the resistance decreases in uniform materials with increasing the wire section and increases proportionally to the wire length.

Moreover, if the wire is not uniform in the longitudinal direction, a local resistivity can be introduced ( r(x), where x is the coordinate along the wire length).

As a function of the local resistivity, the Ohm's law for non homogeneous wires can be written as

V = § r(x) I(x) dx /A

where § represents linear integration.

More complex generalizations can be done for wires that are not uniform also in section and for bodies that are not wire like, where current flows along more complex patterns depending on the application of the potential on the body surface.

Even if these generalizations are out of the scope of a short answer, in any case the linear characteristic of the Ohm's law remains, that is multiplying the applied potential in any point for the same constant, locally the current is also multiplied for the same constant.