Conductivity and Resistivity
Conductivity is the degree to which a specified material conducts electricity, calculated as the ratio of the current density in the material to the electric field that causes the flow of current. Resistivity is the reciprocal of conductivity, and so are interchangeable as long as one tracks the inversions (and, correspondingly, the units). Electrical conductivity tells us how well a material will allow electricity to travel through it, and is similar to thermal conductivity, which tells us the ease upon which thermal energy (heat for most purposes) can move through the material[1].
A conductor is a material which gives very little resistance to the flow of an electric current. Correspondingly, an insulator is a material which is very resistant to the flow of electric current. Semiconductors are materials which display the properties of both conductors and insulators. The most common example of conductive materials are metals, while insulators include materials such as wood or plastics [2]. Semiconductors are generally more rare, but are necessary for the construction of transistors, and as such are present in all computers, with the most common being doped silicon[3]. In addition to these categories, there are superconductors, which have zero resistance under certain conditions (generally extremely low temperatures and/or extremely high pressures)[4], as well as a host of other materials with odd behaviors.
The units for conductivity and resistivity are most naturally expressed in terms of the recognizable Ohm ([math]\displaystyle{ \Omega }[/math]). Resistivity has units of [math]\displaystyle{ \Omega \cdot m }[/math], so conductivity has units of [math]\displaystyle{ \frac{1}{\Omega \cdot m} }[/math]. Expressed in terms of SI base units, the unit of resistivity becomes [math]\displaystyle{ \frac{kg\cdot m^3}{s^3\cdot A^2} = \frac{kg\cdot m^3}{s\cdot C^2} }[/math].
Main Idea
Mathematical Method
Conductivity and Resistivity are properties of a material, dependent primarily on its chemical composition and structure, but also on its temperature and other environmental factors. As such, for our purposes it will always be either a know value, plugged into an equation, or an unknown value, derived from that equation. Our interest is therefore in those equations. The first is on the idealized relation between resistivity and resistance:
[math]\displaystyle{ R = \frac{\rho L}{A} }[/math]
In this, resistance is [math]\displaystyle{ R }[/math], the length of the wire is [math]\displaystyle{ L }[/math], and [math]\displaystyle{ A }[/math] is its cross sectional area. This assumes that there is a clearly defined length (parallel to the direction of current flow) and cross sectional area (perpendicular to the direction of current flow). Naturally, this is exactly the scenario present in a standard wire