Formula of Resistivity: ρ = RA/L, SI Unit Ω m and Examples

The formula of resistivity is $\rho = \frac{RA}{L}$ , where R is the resistance of a wire, A its cross sectional area and L its length. Resistivity strips away the shape of the sample and leaves a number that belongs to the material alone, which is why Indian books also call it specific resistance.

Two copper wires of different thickness have different resistances, yet both have the same resistivity at the same temperature. That makes ρ the right quantity for comparing conductors, insulators and semiconductors. It links straight to the resistance formula and sits in the Class 10 electricity formulas chapter.

Quick facts

Formula
$\rho = \frac{RA}{L}$
Resistance form
$R = \frac{\rho L}{A}$
SI unit
ohm metre (Ω m)
Dimensional formula
[M¹ L³ T⁻³ I⁻²]
Depends on
Material and temperature only
Reciprocal
Conductivity $\sigma = 1/\rho$ , in S/m
Copper
About $1.7 \times 10^{-8}$ Ω m at room temperature

What is the formula of resistivity made of

Quantities in ρ = RA / L
Symbol	Quantity	SI unit
$\rho$	Resistivity, a property of the material (Greek rho)	Ω m
$R$	Measured resistance of the sample	Ω
$A$	Area of the cross section; for a round wire $A = \pi r^2 = \frac{\pi d^2}{4}$	m²
$L$	Length of the sample along the current	m

Resistance rises with length because charge has further to travel through the lattice. It falls with area because a thicker wire offers more parallel paths. Multiplying R by A and dividing by L cancels both effects. The result is the resistance a cube of the material 1 m on each side would show between opposite faces.

Steps to solve resistivity problems

List the data in SI units. Change cm to m, mm to m, and mm² to m² (1 mm² = 10⁻⁶ m²).
Find the area if only a diameter is given. Halve the diameter to get r, then use πr².
Pick the right arrangement. Use ρ = RA/L to find the material value, or R = ρL/A to find the resistance of a given wire.
Check the size of the answer. Metals land near 10⁻⁸ to 10⁻⁶ Ω m. Glass and rubber are above 10¹⁰ Ω m.

Type three of ρ, R, A and L into the solver above and it will work out the missing one.

Resistivity questions worked in full

Question 1: finding the material value

A wire 5.0 m long with a cross section of $1.0 \times 10^{-6}$ m² has a resistance of 2.0 Ω. Find its resistivity.

$\rho = \frac{RA}{L} = \frac{2.0 \times 1.0 \times 10^{-6}}{5.0} = 4.0 \times 10^{-7}\ \Omega\,\text{m}$

This value is typical of an alloy used in heating coils rather than of pure copper.

Question 2: a copper cable with a diameter in millimetres

Copper has $\rho = 1.7 \times 10^{-8}$ Ω m. Find the resistance of 100 m of copper wire of diameter 1.2 mm.

Half the diameter gives r = 0.60 mm = $6.0 \times 10^{-4}$ m, so $A = \pi r^2 = 1.13 \times 10^{-6}$ m².

$R = \frac{\rho L}{A} = \frac{(1.7 \times 10^{-8})(100)}{1.13 \times 10^{-6}} \approx 1.5\ \Omega$

Question 3: stretching a wire

A wire of resistance 4 Ω is stretched to twice its length without change in volume. Find the new resistance.

Volume fixed means the area halves when the length doubles, while ρ is unchanged.

$R' = \frac{\rho (2L)}{A/2} = 4\frac{\rho L}{A} = 4 \times 4 = 16\ \Omega$

The resistivity did not change at all: it is the same metal.

Ohm metre and the dimensions of resistivity

Ohm times square metre divided by metre leaves ohm metre, Ω m. A common slip is writing Ω/m, which is the unit of resistance per length, a different idea.

For dimensions, start from resistance. Since R = V/I and the volt is [M L² T⁻³ I⁻¹], resistance is [M¹ L² T⁻³ I⁻²]. Multiplying by area [L²] and dividing by length [L] adds one power of L:

So resistivity carries [M¹ L³ T⁻³ I⁻²], one length power higher than resistance.

The microscopic form $\rho = \frac{m}{n e^2 \tau}$ , with τ the mean time between collisions, gives the same dimensions and explains why heating a metal raises its resistivity: τ gets shorter. See the drift velocity formula for that model.

Temperature and typical values

For metals, resistivity grows almost in a straight line with temperature: $\rho_T = \rho_0[1 + \alpha(T - T_0)]$ , with α the temperature coefficient. Semiconductors go the other way, because heating frees more charge carriers.

Approximate resistivity near room temperature
Material	ρ (Ω m)	Class
Silver	$1.6 \times 10^{-8}$	Conductor
Copper	$1.7 \times 10^{-8}$	Conductor
Aluminium	$2.7 \times 10^{-8}$	Conductor
Nichrome	$1.1 \times 10^{-6}$	Alloy for heaters
Silicon (pure)	about 10³	Semiconductor
Glass	10¹⁰ to 10¹⁴	Insulator

Alloys like nichrome are chosen for toaster and iron elements because their resistivity is high and changes little with temperature. Copper is used for house wiring because its resistivity is very low.

Mistakes to avoid

Using diameter as radius. The area then comes out four times bigger than it should.
Converting mm² as 10⁻³ m². Square the factor: 1 mm² = (10⁻³ m)² = 10⁻⁶ m².
Expecting ρ to change when a wire is cut or stretched. Only R changes. ρ changes with material or temperature.
Mixing up resistivity and conductivity. σ = 1/ρ, measured in siemens per metre. A good conductor has small ρ and large σ.

Linked idea: current density J and field E obey

E = \rho J

, the local form of Ohm's law. See the current density formula.

Frequently asked questions

What is the formula of resistivity?

Resistivity is ρ = RA / L, resistance times cross sectional area divided by length. Rearranged, the resistance of a wire is R = ρL / A.

Which SI unit does resistivity use?

The ohm metre, written Ω m. It is not ohm per metre. The unit follows from ohm times square metre divided by metre.

Does resistivity depend on length and area?

No. Changing the length or thickness changes the resistance, but the resistivity stays the same because it depends only on the material and its temperature.

How is resistivity different from resistance?

Resistance, in ohms, is a label for one particular wire or resistor, so cutting or thinning it changes the value. Resistivity, in ohm metre, describes the substance itself, whatever shape it is formed into.

Why does the resistivity of a metal increase with temperature?

Hotter ions vibrate more, so free electrons collide with them more often. The mean time between collisions drops, and ρ = m / (ne²τ) goes up.