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Dimensional Formula of Resistance

Dimensional formulas of physical quantitiesClass 10Class 12Updated 1 Nov 2026

[R]=[M1L2T−3I−2][R] = [M^{1} L^{2} T^{-3} I^{-2}]
SI unit
ohm (Ω) = kg m² s⁻³ A⁻²
Dimensions
[M¹ L² T⁻³ I⁻²]
where
  • M dimension of mass kg
  • L dimension of length m
  • T dimension of time s
  • I dimension of electric current A

Working from Ohm's law, the dimensional formula of resistance is [M¹ L² T⁻³ I⁻²]. Current appears as a base quantity alongside mass, length and time, and it carries the power minus two.

Resistance is the ratio of potential difference to current, R = V/I. Voltage is not a base quantity, so the derivation goes through work and charge first. The result looks heavy, but each power has a clear origin. Below are the steps, the ohm in base units, the quantities that share the bracket, a power and time constant check, and the usual slips. Circuit calculations live with the resistance formula and its series and parallel rules.

Quick facts

  • Dimensional formula

    [M¹ L² T⁻³ I⁻²]

  • Defining formula

    R=VIR = \frac{V}{I}

  • SI unit

    ohm (Ω) = V/A

  • In base units

    kg m² s⁻³ A⁻²

  • Scalar or vector

    Scalar

  • Same dimensions as

    Reactance, impedance

Step by step: the dimensional formula of resistance

Build up from mechanics to electricity in four steps.

  1. Charge: q = It, so charge has the bracket [IT].
  2. Potential difference is work per unit charge, V = W/q. Work is [ML²T⁻²]; divide by [IT] to get [ML²T⁻³I⁻¹].
  3. Ohm's law gives R = V/I.
  4. Divide the voltage bracket by [I]: the current power drops from −1 to −2, giving [ML²T⁻³I⁻²].

The T⁻³ comes from T⁻² in work and one more T⁻¹ from the time inside charge. The I⁻² comes from one I in charge and one in the current of Ohm's law. See the voltage and charge pages for those steps on their own.

The ohm, four ways

Spelled out in base units, the bracket is kg m² s⁻³ A⁻². That combination is one ohm (Ω). Grouped more simply, an ohm is a volt per ampere.

Ways to write the ohm
FormWhy it equals one ohm
V/AOhm's law, R = V/I
W/A²From P = I²R, so R = P/I²
J s⁻¹ A⁻²Watt written as joule per second
kg m² s⁻³ A⁻²Base units from the bracket
Each form passes through the same base units. If two of your groupings disagree, one of them has a slip in the power of seconds.

Same bracket and close relatives

Electrical quantities with the bracket [M¹ L² T⁻³ I⁻²]
QuantityRelationComment
ResistanceR = V/IOpposition to steady current
ReactanceX = 1/(ωC) or ωLOpposition from a capacitor or coil in AC
ImpedanceZ = V/I in ACTotal opposition, also in ohms

All three are measured in ohms, so they share one bracket. The impedance formula page combines them. Two nearby brackets to keep apart: resistivity, ρ = RA/l, which multiplies by area and divides by length to give [ML³T⁻³I⁻²] in Ω m, and conductance, G = 1/R, which inverts every power to [M⁻¹L⁻²T³I²] in siemens.

Power and time constant checks

Check: electrical power P = I²R

I² is [I²]. Multiply by the resistance bracket: [I²] × [ML²T⁻³I⁻²] = [ML²T⁻³].

Power, work per second, has exactly this bracket, so the formula is consistent.

Check: the RC time constant has dimensions of time

Capacitance is [M⁻¹L⁻²T⁴I²]. Multiply by resistance: M¹⁻¹, L²⁻², T⁻³⁺⁴, I⁻²⁺² leaves only [T].

So RC is a time, as the charging curve of a capacitor requires. The same check on L/R, with inductance [ML²T⁻²I⁻²], also gives [T].

The capacitance bracket used here is derived on the capacitance page.

Common errors with electrical brackets

  • Writing [ML²T⁻²I⁻²]. Inductance looks like this. Resistance has T⁻³.
  • Writing [ML²T⁻³I⁻¹]. That is voltage. Dividing by current adds one more I⁻¹.
  • Using charge as if it were fundamental. In SI the base quantity is current; charge is [IT].
  • Mixing resistance with resistivity. Resistivity has L³, not L², and the unit Ω m.

Electrical brackets from charge to inductance are gathered with the others under dimensional formulas, ready for revision.

Frequently asked questions

Why does resistance have I⁻² in its dimensional formula?

One I⁻¹ comes from the charge inside voltage, since V = W/q and q = It. The second comes from dividing by current in R = V/I.

What is an ohm made of in base units?

One ohm, the SI unit of resistance, equals kg m² s⁻³ A⁻², or a volt per ampere.

Do resistance and impedance share a bracket?

They do: each is a voltage over a current, measured in ohms, so both are [ML²T⁻³I⁻²].

How are the dimensions of resistance and resistivity different?

Resistivity is resistance times area over length, so it has one more L. Resistance is [ML²T⁻³I⁻²]; resistivity is [ML³T⁻³I⁻²].

Is resistance a scalar?

Yes. Resistance has a size in ohms and no direction.