Resistivity Formula

This article discusses the resistivity formula and its derivation. Resistivity refers to the electrical resistance of a conductor of a particular unit cross-sectional area and unit length. It is definitely a characteristic property of each material. Furthermore, experts can use resistivity for comparing different materials on the basis of their ability to conduct electric currents. High resistivity is the designation of poor conductors.

What is Resistivity?

Resistivity or electric resistivity is certainly the inverse of the electrical conductivity. Resistivity is a fundamental property of a material and it demonstrates how strongly the material resists or conducts electric current. A low resistivity is a clear indication of a material which readily allows electric current. Moreover, the common representation of resistivity is by the Greek letter \(\rho\). Also, the SI unit of electrical resistivity happens to be the ohm-meter (\(\rho\)-m).

Resistivity is certainly the measure of how strongly a particular material opposes the flow of electric current on conductors or resistors with a certain uniform cross-section. Furthermore, uniform cross-section is the one where the current flows in a uniform manner. Conductivity happens to be the reciprocal quantity which is a measure of the easiness by which a material permits the flow of current.

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Resistivity Formula

The resistivity formula can be represented as follows:

Resistivity = \(\frac{1}{conductivity}\)

The equation can be represented as :

\(\rho\) = \(\frac{1}{\sigma }\)

Here:

\(\sigma\) = conductivity
\(\rho\) = resistivity

Furthermore, another formula for resistivity can be

\(\rho\) = \(\frac{RA}{L}\)

Here,

\(\rho\) = resistance
R = resistance
A = cross-sectional
L = length

Resistivity Formula Derivation

The resistance R is definitely directly proportional to the length of the conductor. What this reflects is that resistance increases with an increase in the conductor’s length.

So, resistance (R) \(\propto\) l (1)

The resistance R is certainly inversely proportional to the area of cross-section of a particular conductor. What this means is that R will decrease with an increase in the area of conductor and vice-versa. More area of the conductor leads to an efficient flow of electric current through more area and consequently decreases the resistance.

Therefore, resistance \(\propto\) \(\frac{1}{A}\) of cross section of the conductor (A)

Or, R \(\propto\) \(\frac{1}{A}\) (2)

Now from equation (1) and (2)

R \(\propto\) \(\frac{l}{A}\)

Or, R = \(p\frac{l}{A}\) (3)

Here \(\rho\) (rho) happens to be the proportionality constant. Most noteworthy, it is the electrical resistivity of the material of conductors.

Now from equation (3)

RA = \(\rho\)l

Or its alternate can be, \(\rho\) = \(\frac{RA}{l}\)

Solved Examples on Resistivity Formula

Q1 Find out the resistivity of the metal wire of 2 m length and 0.6 mm in the diameter if its resistance happens to be 50 Ω

A1 The given information contains:

Resistance (R) = 50 Ω
Length (l) = 2 m
Diameter = 0.6 mm
Therefore, the radius would be 0.3 mm = 3 × 10-4m
Resistivity (\(\rho\)) = ?

The area of the cross section of wire is = \(\pi r^{2}\)
Or it can be A = 3.14 × (3×10-4)2

Also, A = 28.26 × 10-8 m2 = 2.826 × 10-9 m2

It is already known that

\(\rho\) = \(\frac{RA}{l}\)
Or, \(\rho\) = \(\frac{50\Omega \times 2.826\times 10^{-9}m^{2}}{2m}\)
\(\rho\) = 25 × 2.826 × 10-9Ωm
= 70.65 × 10-9Ωm

Finally, \(\rho\) = 7.065 × 10-8Ωm