What is Resistivity : Definition and Its Formula

When a potential difference is applied across a material, the electrons in the material start moving from the negative electrode to positive electrodes, which produces current in the material. But during this movement of electrons, they undergo various collisions with other electrons in their path. These collisions cause some opposition to the flow of electrons. This phenomenon is known as Resistance to the material. Resistivity property of the materials is beneficial in electrical circuits. Many factors affect the resistance value of a material. The value of specific resistance of the material gives us an idea about the resistive capacity of a particular material.

What is Resistivity?

Materials are divided based on their conducting properties as conductors, semiconductors, and insulators. The electrical resistivity of a material is defined as the resistance of the material per unit length and per unit cross-sectional area at a specified temperature.

When a potential difference is applied across a substance, the resistance property of the substance opposes the flow of current through it. This property of the substance varies with the temperature and also depends on the type of material the substance is made up of. it measures the resistance of the substance.

Formula for Resistivity

The formula for this is derived from the laws of resistance. There are four laws for the resistance of a substance.

First Law

It states that the resistance of a substance R is directly proportional to its length L. i.e. R ∝ L. Thus when the length of the substance is doubled. its resistance also gets doubled.

Second Law

According to this law, the resistance R of a substance is indirectly proportional to its cross-sectional area A. i.e. R ∝ 1/A. Thus by doubling the cross-sectional area of a substance, its resistance value is halved.

Third Law

This law states that the resistance of a material depends on the temperature.

Fourth Law

According to this law, the resistance value of two-wire made up of different materials is different although they are the same in their length and cross-sectional areas.

From all these laws the resistance value of a conductor with  length L and cross-sectional area A can be derived as

R ∝ L/A

R = ρL/A

Here, ρ is the resistance co-efficient known as Resistivity of specific resistance.

Thus the electrical resistivity of the material is given as

ρ = RA/L

The S.I unit of its is Ohm-Meter. It is denoted by the symbol ‘ρ’.

Resistivity Classification for Conductors, Semiconductors, and Insulators

This material highly depends on the temperature. In conductors with the increase in temperature the speed of electrons moving in the material also increases. This leads to a lot of collisions. This results in a decrease in the average time of collision of the electrons. This substance is inversely proportional to the average time of the collision of electrons. Thus, with the decrease in the average time of the collision, the resistivity value of the conductor increases.

In semiconductor substances when the temperature is increased the breaking of more covalent bonds occurs. This increases the number of free charge carriers in the substance. With this increase in charge carriers, the conductivity of the substance increases thereby decreasing the resistivity of the semiconductor material. Thus with the increase in temperature, its semiconductors will increase.

it helps in comparing the various materials based on their ability to conduct electricity. it is reciprocal of conductivity. Conductors have high conductivity values and lower resistivity values. Insulators have high resistivity values and low conductivity values. The values of resistivity and conductivity for semiconductor lies in the middle.

Its value for a good conductor such as Hand-drawn copper at 20⁰ C is 1.77 ×10^-8 ohm-meter and on the other hand, this for a good insulator ranges from 10¹² to 10²⁰ ohm-meters.

Temperature Coefficient

The temperature coefficient of resistance is defined as the change in the increase in the resistance of 1Ω resistor of a material per 1⁰ C rise in the temperature. It is denoted by the symbol ‘α’.

The change in the resistivity of the material with the change in temperature is given as

dρ/dt = ρ. α

Here, dρ is the change in the resistivity value. Its units are ohm-m² /m. ‘ρ’ is the resistivity value of the substance. ‘dt’ is the change in temperature value. ‘α’ is the temperature coefficient of resistance.

The new resistivity value for material when it undergoes temperature change can be calculated by the above equation. First, the amount of change in its value is calculated using the temperature coefficient. Then the value is added to the previous value to calculate the new value.

This is very useful in calculating the resistance values of the material at various temperatures. Resistance and resistivity both terms are related to the opposition experienced by a flowing current but it is an intrinsic property of the materials. All the copper wires irrespective of their length and cross-sectional area have the same resistivity value whereas their resistance value changes with change in their length and cross-sectional areas.

Every material has its value. The general resistivity values for different types of material can be given as – For superconductors resistivity is 0, for metals resistivity is 10^-8, for semiconductors and electrolytes resistivity value is variable, for insulators resistivity value is from 10¹⁶, for super insulators the resistivity value is ‘∞’.

At 20⁰ C the resistivity value for silver is 1.59×10^-8, for copper 1.68×10-8. All the resistivity values for various materials can be found in a table. Wood is considered as a high-insulator but this varies depending upon the amount of moisture present in it. In many cases, it is difficult to calculate the resistance of a material using the resistivity formula due to the inhomogeneous nature of the materials. In such cases, the partial differential equation formed by the continuity equation of J and Poisson’s equation for E is used. Do the two wires with different lengths and different cross-sectional areas have the same values?