# Definition of Conductivity: A Deep Dive

## I. Resistivity and Conductivity

The correct term is conductivity, measured in Siemens, where one Siemens equals the reciprocal of one Ohm. The reciprocal of resistivity (1/ρ) is termed as conductivity, symbolized as σ, denoted in Siemens per meter (S/m).

Conductance (G) is measured in Siemens (S), where 1S equals the reciprocal of one Ohm (1/1Ω).

1. Defining Resistivity

Resistivity is a physical quantity used to represent the electrical resistance characteristics of various substances.

The resistance of a conductor that is one meter in length and has a cross-sectional area of one square millimeter, made from a certain material, is referred to as the resistivity of that material.

2. Units of Resistivity

In the International System of Units, the unit of resistivity is Ohm-meter. A commonly used unit is Ohm-square millimeter/meter.

3. Factors Affecting Resistivity

Resistivity (ρ) depends not only on the material of the conductor but also on the temperature of the conductor.

Within a relatively stable temperature range, the resistivity of almost all metals linearly changes with temperature, i.e., ρ=ρo(1+at).

In this formula, t is the Celsius temperature, ρo is the resistivity at 0℃, and a is the temperature coefficient of resistivity.

Due to the change in resistivity with temperature, the resistance of some electrical appliances must be stated under specific physical conditions.

For example, a 220V 100W light bulb filament has a resistance of 484 Ohms when energized, but only around 40 Ohms when not energized.

4. Distinction between Resistivity and Resistance

Resistivity and resistance are two different concepts. Resistivity reflects the property of a material to resist current flow, while resistance reflects the resistance to current flow by an object.

5. Resistivity of Various Metal Conductors at 20℃

As can be seen, metals have lower resistivity, alloys have higher resistivity, and non-metals along with some metal oxides have even higher resistivity. Insulators have extremely high resistivity.

Germanium, silicon, selenium, cupric oxide, boron, and similar materials have lower resistivity than insulators but higher than metals, and we categorize these materials as semiconductors.

### 5. Conclusion

Under normal conditions (as can be seen from the table), the best conductors in order are silver, copper, and aluminum. These three materials are the most commonly used and are often used as conductors.

Copper is the most widely used, and almost all current conductors are made of copper (except for precision instruments and special occasions).

Aluminum wire has been phased out due to its chemical instability and propensity to oxidize.

Silver has the best conductivity, but due to its high cost, it is rarely used and is only used in high-demand situations such as precision instruments, high-frequency oscillators, and aerospace applications.

Gold is also used for contacts on some instrumentations due to its chemical stability, not because of its low resistivity.

## II. Understanding Resistance and Ohm’s Law

German physicist Georg Simon Ohm (1789-1854) summarized over several years from 1825 to 1827 what we now call Ohm’s Law. Although the concepts of current and voltage were not precise at the time, the measuring tools were designed and manufactured by Ohm himself.

The power source initially used was a primitive Voltaic pile, which was later replaced by a thermocouple. The conductors selected were metal wires of various lengths and thicknesses.

Even though there were sporadic mentions of “conductance,” the concept of “resistance” as a physical quantity had not yet formed.

Under such conditions, Ohm established the relationship between current and voltage through experimentation and defined resistance.

Understanding this historical context helps us fully comprehend resistance and Ohm’s Law today.

In modern terms, the result of Ohm’s experiment is that the current I passing through a conductor (metal wire) is directly proportional to the voltage U at its ends, i.e.,

I ∝ U (1)

Expressed as an equation,

R = U/I (2)

Here, R is a constant of proportionality, a quantity that doesn’t vary with U and I, or is independent of U and I. R is determined by the conductor, reflecting the conductor’s property of impeding the current, known as resistance, which is defined as

R = U/I (3)

After establishing the direct proportionality of current to voltage and defining resistance, we arrive at the current form of Ohm’s Law:

I = U/R (4)

In the history of physics, the establishment of laws in each new field often involves both experimental results and the definition of new physical quantities. The establishment of Ohm’s Law is no exception.

Hence, Ohm’s Law serves as both a circuit law and a description of the properties of conductors.

It’s referred to as an electrical law because it describes the relationship among current, voltage, and resistance in a steady-state circuit. It’s also a description of the conductor’s properties, as it includes the definition of resistance. The two aspects cannot be separated or opposed.

From the formulation of Ohm’s law, the statement “resistance is independent of voltage and current” refers to the resistance defined as a constant ratio, which does not change with variations in voltage and current.

The resistance of a conductor is an inherent property, determined by the conductor itself. Equation (5), often called “Ohm’s law,” stipulates that resistance is determined by the length L of the conductor, its cross-sectional area S, and the properties ρ of the material forming the conductor:

R = ρ·L/S (5)

It is independent of the voltage across the conductor and the current passing through it.

The statement “resistance is independent of voltage and current” refers to the determinant of resistance, meaning that voltage and current are not the factors determining resistance.

This aligns with the definition of resistance as a constant ratio, unaltered by changes in voltage and current.

Therefore, the assertion “resistance is independent of voltage and current” has its prerequisites and applicable scope and should not be taken as absolute!

By the way, it’s important not to absolutize when examining specific issues. This applies to questions related to resistance and Ohm’s law.

The material property represented by ρ is called “resistivity.” Resistivity ρ is related to temperature t.

Generally, the resistivity ρ of a metal increases as the temperature t rises. If ρ0 is the resistivity at 0°C, then the resistivity at temperature t is:

ρ = ρ0(1+αt) (6).

Here α is the “temperature coefficient of resistance”. Pure metals generally have a high α, with copper at 0.0039°C-1, tungsten at 0.0045°C-1, iron at 0.005°C-1, and nickel at 0.006°C-1.

This means that for every 100°C rise in temperature, resistivity increases by about 40% to 60%.

However, some alloys have very low α values, like Manganin (84% copper, 12% manganese, 4% nickel) at 0.00001°C-1, and Constantan (54% copper, 46% nickel) at 0.00004°C-1, showing almost no change with temperature.

Interestingly, the resistivity of electrolytes (aqueous solutions of acids, bases, salts) and some materials like carbon decreases as temperature rises. This shows that the change in resistivity with temperature should not be absolutized either.