Dimensional Formula of Surface Tension [MT⁻²] with Derivation

Expressed in base quantities, the dimensional formula of surface tension is [M¹ L⁰ T⁻²], usually written [MT⁻²]. Length cancels out completely, which surprises many students the first time they see it.

Surface tension is force per unit length along a line drawn on the liquid surface, S = F/l. Force has one power of length, the line length removes it, and only mass and time remain. The same bracket comes from surface energy per unit area. This page covers both routes, the unit N/m, how surface tension differs from ordinary tension, a capillary rise check and the slips. For worked liquid problems, go to the surface tension formula.

Quick facts

Dimensional formula
[M¹ L⁰ T⁻²], short form [MT⁻²]
Defining formula
$S = \frac{F}{l}$
SI unit
N/m, equal to J/m²
CGS unit
dyne/cm; 1 N/m = 1000 dyne/cm
Same dimensions as
Surface energy, spring constant
Chapter
Mechanical properties of fluids, Class 11

Two derivations of the dimensional formula of surface tension

Route 1: force per length

$S = \frac{F}{l}$ .
$[F] = [M L T^{-2}]$ and $[l] = [L]$ .
Divide: the single L cancels, leaving $[M T^{-2}]$ , or $[M^{1} L^{0} T^{-2}]$ in full.

Route 2: energy per area

Surface energy per unit area: $S = \frac{W}{A}$ .
$[W] = [M L^{2} T^{-2}]$ and $[A] = [L^{2}]$ .
The two powers of L cancel, and once more only $[M T^{-2}]$ is left.

Two routes, one bracket. That is why N/m and J/m² are the same unit: multiply the top and bottom of N/m by a metre and you get N m per m², a joule per square metre.

N/m, J/m² and dyne/cm

Reading [MT⁻²] as base units gives kg s⁻². Written in the usual way, the SI unit is the newton per metre, N/m, equal to J/m².

Convert the surface tension of water, 0.072 N/m, to dyne/cm

One newton is 10⁵ dyne and one metre is 10² cm.

$0.072\ \frac{\text{N}}{\text{m}} = 0.072 \times \frac{10^{5}\ \text{dyne}}{10^{2}\ \text{cm}} = 72\ \text{dyne/cm}$ .

The factor is 1000. You can see it in the dimensions too: [MT⁻²] has only mass, so only the kg to g factor, 10³, survives.

Tension, surface tension and other matches

Students often search for the dimensional formula of tension and surface tension together, because the names are so close. The quantities are not the same.

Tension versus surface tension
Quantity	What it is	Dimensions	SI unit
Tension in a string	A force	[MLT⁻²]	N
Surface tension	Force per length	[MT⁻²]	N/m

Other quantities that share [MT⁻²]:

Same bracket as surface tension
Quantity	Relation
Surface energy per unit area	$\frac{W}{A}$
Spring constant	$k = \frac{F}{x}$
Any force per unit length	$\frac{F}{l}$

The spring constant match is a favorite one mark question. A spring's k and a liquid's S have unrelated physics but both are force divided by a length. See spring constant and tension.

Capillary rise as a dimension check

Check the capillary rise formula h = 2S cos θ/(ρgr)

Top: S is [MT⁻²] and cos θ is a number. Bottom: ρgr has [ML⁻³] × [LT⁻²] × [L], which is [ML⁻¹T⁻²].

Ratio: [MT⁻²] ÷ [ML⁻¹T⁻²] leaves [L], a height. The formula is consistent.

Numbers: in a clean glass tube 0.25 mm in radius, with θ = 0, water climbs h = 2 × 0.072/(1000 × 9.8 × 2.5 × 10⁻⁴) = 0.059 m, roughly 5.9 cm.

A second check: the excess pressure inside a drop, ΔP = 2S/R, is [MT⁻²]/[L] = [ML⁻¹T⁻²], a pressure, as it must be.

Mistakes in fluid dimension questions

Writing [MLT⁻²]. That is tension, a force. Surface tension divides by length.
Writing [ML⁻¹T⁻²]. That is pressure, force per area. Surface tension is per length only.
Expecting a length in the bracket. L⁰ is correct; the length powers cancel.
Converting N/m to dyne/cm by 10⁵. The length factor 10² divides it down to 10³.

Fluid quantities such as viscosity and pressure sit next to this one on the dimensional formulas page.

Frequently asked questions

Why has surface tension no length in its dimensional formula?

Force contributes L¹ and the length of the line contributes L⁻¹. They cancel, leaving [MT⁻²].

What unit is surface tension measured in?

Newton per metre, N/m, which is the same as joule per square metre. In CGS it is dyne/cm, and 1 N/m = 1000 dyne/cm.

Do surface tension and surface energy have the same dimensions?

Yes. Surface energy per unit area is J/m², and J/m² equals N/m, so both are [MT⁻²].

Does surface tension have a direction?

Surface tension itself is treated as a scalar property of the liquid. The force it produces acts along the surface, perpendicular to any line drawn on it.

Which quantity has the same dimensions as surface tension?

The spring constant k. Both are force divided by length, so both are [MT⁻²].