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 the two 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 with which thermal energy (heat for most purposes) can move through the material.

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 . 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. In addition to these categories, there are superconductors, which have zero resistance under certain conditions (generally extremely low temperatures and/or extremely high pressures), and there also exist 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

First, we have the relation between conductivity and resistivity:

[math]\displaystyle{ \sigma = \frac{1}{\rho} }[/math]

By convention, we have [math]\displaystyle{ \sigma }[/math] as the conductivity, and [math]\displaystyle{ \rho }[/math] as the resistivity.

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. The next two relevant equations are two statements of the same fact, which we call Ohm's law. Precisely, it governs the relationship between the flow of charge and the electric field which produces that flow. The first form states this explicitly:

[math]\displaystyle{ \vec{J} = \sigma \vec{E} }[/math]

Here [math]\displaystyle{ \vec{J} }[/math] is the current density, and [math]\displaystyle{ \vec{E} }[/math] is the electric field as we've seen before. An aside on current density: it is the amount of charge which passes through a given cross sectional area in a given period of time, with SI units [math]\displaystyle{ \frac{A}{m^2} = \frac{C}{m^2 \cdot s} }[/math]. This means that it is a density with area in the denominator, which can be confusing. Given what one has learned so far, this is the operable definition of Ohm's Law. However, another version is more frequently used later on, and so will also be given here:

[math]\displaystyle{ V = I R }[/math]

where [math]\displaystyle{ V }[/math] is the electric potential, [math]\displaystyle{ I }[/math] is the current, and [math]\displaystyle{ R }[/math] is resistance as defined before.

Computational Method

This topic is largely conceptual and algebraic, so there is relatively little modelling to be done. However, this program is designed to create plots of the relationship between resistance, resistivity, length and cross sectional area. Furthermore, this external program provides a good visual demonstration of the relationship (but unfortunately the source code is not accessible).

Examples

Simple

An material has a resistivity of [math]\displaystyle{ 200 \; \Omega\cdot m }[/math]. What is its conductivity?

Solution

Since conductivity is the reciprocal of resistivity,

[math]\displaystyle{ \sigma = \frac{1}{\rho} = \frac{1}{200 \; \Omega \cdot m} = 0.005 \frac{1}{\Omega \cdot m} }[/math]

Middling

An electric potential of [math]\displaystyle{ 120 V }[/math] is applied to a circular wire of length [math]\displaystyle{ 2 \cdot 10^4 \; m }[/math] and radius [math]\displaystyle{ 0.001 m }[/math]. The current is equal to [math]\displaystyle{ 1.11 \; A }[/math]. Determine the resistivity, and match it to an elemental metal using an appropriate table (such as )

Solution

We have Ohm's Law

[math]\displaystyle{ V = I R }[/math]

and so plugging in our definition for resistance in terms of resistivity gives that

[math]\displaystyle{ V = \frac{I \rho L}{A} }[/math]

which we rearrange to get

[math]\displaystyle{ \rho = \frac{ V A}{I L} }[/math]

plugging all of the values in gives an answer of [math]\displaystyle{ \rho = 1.7 \cdot 10^{-8} \; \Omega \cdot m }[/math], which is the resistivity of copper.

Difficult

One cubic meter of a fictional material with resistivity [math]\displaystyle{ \rho = 10^{-5} \; \Omega \cdot m }[/math] is formed into a shape with uniform cross sectional area (such that volume is equal to the base times the height) for which the resistance to current run lengthwise is equal to [math]\displaystyle{ 10 \; \Omega }[/math]. Determine the dimensions (length and cross sectional area) of the shape, presuming that it follows Ohm's law and the equation for resistance given above.

Solution

As the clarification hints, it is necessary to use a little basic geometry to solve this problem, namely

[math]\displaystyle{ V = L \cdot A }[/math]

So we lay out the standard formula

[math]\displaystyle{ R = \frac{\rho L}{A} }[/math]

Then multiply both numerator and denominator by [math]\displaystyle{ L }[/math] to obtain

[math]\displaystyle{ R = \frac{\rho L^2}{A L} = \frac{\rho L^2}{V} }[/math]

From here it is simply rearrangement to find that

[math]\displaystyle{ L = \sqrt{\frac{R V}{\rho}} = \sqrt{\frac{(1 \;m^3)(10 \;\Omega)}{10^{-5} \; \Omega \cdot m }} = 10^2 \; m }[/math]

With the length determined, it is then straightforward to conclude that [math]\displaystyle{ A = 10^{-2} \; m^2 }[/math], and the cross-section can take any shape which has that enclosed area.

Connectedness

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History

The first scientist to contribute substantially to the study of conductivity and resistivity was Stephen Gray, who with his friend Ganvil Wheler noticed that electricity could be transmitted a distance, and that the effectiveness of this transmission varied between a wire of silk and wire of brass . A substantial amount of discovery then followed, including the famous contributions of Benjamin Franklin, but these developments are more relevant to other subjects. It was in the early 19th century that the equation for determining resistance from resistivity and geometry was determined by Antoine Becquerel (grandfather to Henri Becquerel, who was a pioneer in radioactivity alongside Marie Skłodowska Curie and Pierre Curie). Not long after, Georg Ohm proposed an incorrect theorem for the relation between current, potential and resistance, then soon amended it to the correct formula. However, the correction, philosophical differences, and confusion at the time about the definitions of resistance, current and potential led his work to be poorly received at first, but ultimately accepted and celebrated about 15 years after its release.