Formula of Resistance: R = V/I and R = ρL/A, Examples

The formula of resistance is $R = \frac{V}{I}$ : the potential difference across a conductor divided by the current it carries. One ohm is the resistance that lets one ampere flow when one volt is applied.

A second formula tells you what sets that value for a wire: $R = \frac{\rho L}{A}$ , where ρ is the resistivity of the material, L the length and A the cross section area. Class 10 students learn both, and they return in Class 12, AP Physics 1 and every circuits chapter after that.

Quick facts

Definition
$R = \frac{V}{I}$
For a wire
$R = \frac{\rho L}{A}$
SI unit
ohm (Ω) = V/A
Dimensional formula
[M¹ L² T⁻³ I⁻²]
Series
$R = R_1 + R_2 + R_3$
Parallel
$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}$
Type
Scalar, always positive for ordinary resistors

Two forms of the formula of resistance in physics

Symbols in R = V / I and R = ρL / A
Symbol	Quantity	SI unit
$R$	Resistance	Ω
$V$	Potential difference	V
$I$	Current	A
$\rho$	Resistivity of the material	Ω m
$L$	Length of the conductor	m
$A$	Area of cross section	m²

The first form defines resistance for any component. The second predicts it from shape and material. Resistance doubles when you double the length and halves when you double the area. The material enters through ρ, explained on the resistivity formula page.

Three resistance calculations

Example 1: a lamp filament

A lamp on a 230 V supply draws 0.50 A. Find the resistance of its hot filament.

$R = \frac{230}{0.50} = 460\ \Omega$

Example 2: a copper wire with area in mm²

Find the resistance of 20 m of copper wire with cross section 0.50 mm². Resistivity of copper is 1.7 × 10⁻⁸ Ω m.

A = 0.50 × 10⁻⁶ m².

$R = \frac{(1.7\times10^{-8})(20)}{0.50\times10^{-6}} = 0.68\ \Omega$

Example 3: series and parallel

Resistors of 6 Ω and 3 Ω are joined first in series, then in parallel. Find each combined resistance.

Series: R = 6 + 3 = 9 Ω.

$\frac{1}{R} = \frac{1}{6} + \frac{1}{3} = \frac{1}{2}\quad\Rightarrow\quad R = 2\ \Omega$

The parallel value is always smaller than the smallest single resistor.

What changes resistance

Factors and their effects
Change	Effect on R
Double the length	R doubles
Double the cross section area	R halves
Double the diameter	Area is 4 times, so R becomes one quarter
Stretch a wire to twice its length (same volume)	Area halves too, so R becomes 4 times
Heat a metal	R increases

For metals, a useful approximation is $R_T = R_0(1 + \alpha\,\Delta T)$ , where α is the temperature coefficient of resistance. Semiconductors behave the other way: their resistance falls as they warm up.

The stretched wire case is a favourite objective question. Volume stays fixed, so if L becomes nL, A becomes A/n and R becomes n²R.

Units and dimensions of the ohm

Since 1 V = 1 W/A and 1 W = 1 J/s, the ohm in base units is kg m² s⁻³ A⁻². That gives the bracket form [M¹ L² T⁻³ I⁻²], derived step by step on the dimensional formula of resistance page. Large values are written in kΩ (10³ Ω) and MΩ (10⁶ Ω). The reciprocal of resistance is conductance, measured in siemens (S).

Resistance, Ohm's law and power

Ohm's law says that for many conductors at steady temperature V is proportional to I, so R is constant: the V against I graph rises as a straight line from zero, and its gradient equals R. The definition R = V/I still holds for a lamp or a diode, but there R changes with current, and the graph curves. The Ohm's law page covers this distinction.

Resistance also controls heating. The power used is $P = I^2R = \frac{V^2}{R}$ , which is why a heater element has a lower resistance than a bulb on the same supply.

Mistakes that lose marks

Leaving the area in mm² or cm² in R = ρL/A.
Adding parallel resistors directly, or forgetting to invert 1/R at the end.
Using the diameter as the radius.
Using mA without converting to A before dividing.

The electricity formula hub holds every formula for the Class 10 chapter, with separate pages for voltage and current.

Frequently asked questions

What is the formula of resistance?

R = V / I. For a uniform wire, R = ρL / A, where ρ is resistivity, L is length and A is cross section area.

What do we measure resistance in?

The ohm (Ω). One ohm equals one volt per ampere.

What is the formula for resistances in series and parallel?

In series, add them: R = R₁ + R₂ + R₃. In parallel, add the reciprocals: 1/R = 1/R₁ + 1/R₂ + 1/R₃.

Does resistance depend on voltage?

For an ohmic conductor at constant temperature, no. Changing the voltage changes the current in the same ratio, so V/I stays the same.

What is the difference between resistance and resistivity?

Resistance belongs to a particular object and depends on its size and shape. Resistivity is a property of the material alone, in Ω m.