Understanding Resistivity: Meaning, Principles, Formula, and Calculations

This article will clearly explain the concept of electrical resistivity in materials. In other terms, it is also known as specific resistance. In electrical engineering, every material offers a finite amount of resistance to the flow of electric current. However, this resistance is not the same for all materials—it varies depending on their dimensions and intrinsic properties. For example, a material like copper will exhibit different resistance values when its dimensions or physical properties change.

However, each material has a constant value of resistance per unit dimension when its properties remain unchanged. Let us explore this topic in more detail

It is very important to understand the concept of resistance before studying resistivity (also known as specific resistance). Resistance is a property of a material that describes its ability to restrict the flow of electric current. Materials with higher resistance offer greater opposition to current flow, while materials with lower resistance allow current to pass more easily.

In electrical engineering, materials with very low resistance are used as conductors, while those with high resistance are used as insulators. However, the amount of resistance offered by a material depends on both its dimensions and inherent properties.

This brings us to the concept of resistivity, which is the resistance of a material per unit length and unit cross-sectional area, assuming its properties remain constant. In other words, resistivity is a measure of how strongly a material opposes the flow of current, independent of its shape or size.

The specific resistance of a material is defined as the resistance measured between two opposite ends of a material with unit length and unit cross-sectional area, assuming its properties remain unchanged.

In other words, resistivity is the resistance of a cubic material that has a length of 1 cm and a cross-sectional area of 1 cm², while its material properties remain constant.

With reference to the above figure, a cubic material is used to conduct an experiment to determine its resistivity. It has a length of 1 cm and a cross-sectional area of 1 cm², and its properties are assumed to remain constant. An ohmmeter is connected across its two opposite ends. The value of resistance indicated by the ohmmeter is known as the specific resistance (or resistivity) of the material. This is the base resistance, which will change if the material’s dimensions change

Resistivity provides information about the base resistance of a material. Although resistivity itself is a material property and remains constant under fixed conditions, the measured resistance changes when the dimensions of the material change.

• When the length of a material increases, its resistance increases. Similarly, when the length decreases, the resistance decreases. This is because the electric current has to travel a longer path to exit the conductor.
• When the cross-sectional area of a material increases, the resistance decreases. Conversely, when the area decreases, the resistance increases. A larger area allows the current to pass more easily.

To visualize this, imagine walking through a narrow corridor versus an open space. A corridor typically has less area and more length, so it takes more time to pass through. In contrast, an open space offers more area and usually a shorter path, making it easier to move through. The same principle applies to the flow of electric current in a material.

To represent it in mathematically

R α l

R α 1/a

Combining both of above equation

R = l/a

By replacing the proportionality constant (α) with ρ, which represents the resistivity of the material

R = ρ x (l/a)

Here,

R = Resistance, measured in Ohms (Ω)

L = Length of Material, in cm

A = Cross-sectional of material in cm²

ρ = Resistivity of material

This equation clearly indicates that the resistance of a specimen material is the product of its resistivity (ρ) and the ratio of its length (L) to its cross-sectional area (A).

The equation for the resistivity of a specimen material can be derived from the equation for the resistance of the material

R = ρ x l/a

By re-writing it, we obtain following equation

ρ = R x a/l

The unit of resistivity can be obtained by substituting the units of the respective parameters into the resistivity equation

ρ = (Ohm) x (cm²) /cm

Simplifying it, we get unit of resistivity

ρ = (Ohm) x (cm)

So resistivity (or) Specific resistance unit known as Ohm-cm

Material Name	Specific Resistance in Ohm -Cm
Copper	1.7 x 10-8
Iron	9.68 x 10 -8
Nichrome	100 x 10 -8
Glass	1010 to 1014
Rubber	1013
Silver	1.6 x10-8
Aluminium	2.65 x 10-8
Tungsten	5.6 x 10-8
Constantan	4.9 x 10-7
Mica	1011 to 1015

Problem:1

Find out the resistance of conductor having 120 m length of wire having uniform cross sectional area of 0.02 mm² and having resistivity of 50 uΩ-cm. If wire pulled thrice of its length from its original length , Calculate the resistance value

GIven:

Length of wire (l) = 120 m

Cross sectional of wire (a) = 0.02 mm²

Resistivity of wire (ρ) = 50 uΩ-cm.

To find:

Resistance of wire

1. When it is original length
2. When it pulled out to thrice of its length

Solution

1. When it is original length

Resistance R = ρ x l/a

Substituting given values in above resistance equation

R = 50 x 10^-6 x ((120 x 10²)/(0.02 x 10^-2 ))

R = 3000 Ω

2. When it pulled out to thrice of its length

New Length of wire l₂ = 3 x l = 3l

But it is important to note , that when wire pulled to thrice of its length its cross sectional area will be decreased. Because volume of wire remains constant

Volume (v) = cross-sectional area (a) x Length (l)

a x 1 = a₂ x l₂

a₂ = (a xl) / l₂

New cross sectional area of wire a₂ = (a xl) / l₂

where l =1 and l₂ = 3l

a₂ = (a x1) / 3

a₂ = a / 3

a2 = (0.02 x10 ^-2) / 3

0.00006 mm²

New Resistance (R₂) = l₂/a₂

R₂ = 50 x 10^-6 x ((360 x 10²)/(0.00006))

R₂ = 27000 Ohm

Result

1. Resistance when wire it is in original length = 3000 Ohm
2. When it stretched to thrice = 27000 Ohm