Permittivity of Free Space Dimensional Formula [M⁻¹L⁻³T⁴I²]

Written with current as the fourth base quantity, the permittivity of free space dimensional formula comes out as [M⁻¹ L⁻³ T⁴ I²]. It is one of the few brackets in Class 12 physics where mass carries a negative power and time a power as high as four.

The constant ε₀ (epsilon nought) appears in Coulomb's law, where it sets how strongly two charges push or pull through a vacuum. Its value is 8.854 × 10⁻¹² C²/(N m²). Because charge is not a base quantity in SI, the derivation first rewrites charge as current times time. Below you will find that working, the unit F/m, a check with the speed of light and capacitance, and the slips. The force law itself is on the electrostatic force formula page.

Quick facts

Dimensional formula
[M⁻¹ L⁻³ T⁴ I²]
Defining law
Coulomb's law, $F = \frac{q_1 q_2}{4\pi\varepsilon_0 r^{2}}$
Value
8.854 × 10⁻¹² C²/(N m²)
SI unit
C²/(N m²) = F/m
Related constant
1/(4πε₀) = 8.99 × 10⁹ N m²/C²
Chapter
Electric charges and fields, Class 12

Deriving the permittivity of free space dimensional formula

Coulomb\'s law: $F = \frac{q_1 q_2}{4\pi\varepsilon_0 r^{2}}$ , so $\varepsilon_0 = \frac{q_1 q_2}{4\pi F r^{2}}$ .
Since a charge equals a current flowing for a time (q = It), each charge is [IT] and the product q₁q₂ is [I²T²].
The bottom: force [MLT⁻²] times r² [L²] gives [ML³T⁻²]; 4π has no dimensions.
Divide: [I²T²] ÷ [ML³T⁻²]. Mass goes to M⁻¹, length to L⁻³, time to T²⁺² = T⁴, current stays I².

That completes the bracket for ε₀. The step that trips most students is the time power: T² from the charges and T⁺² from flipping the T⁻² of force add to T⁴. For charge alone, see the charge derivation.

Three ways to write the unit

Converted letter by letter, ε₀ is measured in kg⁻¹ m⁻³ s⁴ A². Few people write it like that; it is usually grouped in one of two tidier ways.

Equivalent units for ε₀
Unit	Where it comes from
C²/(N m²)	Rearranging Coulomb's law
F/m (farad per metre)	From C = ε₀A/d for a capacitor
kg⁻¹ m⁻³ s⁴ A²	Straight from the dimensions

All three are the same unit. F/m is the neatest and is what most data tables print. The constant 1/(4πε₀) = 8.99 × 10⁹ N m²/C² has the inverse bracket, [M L³ T⁻⁴ I⁻²].

Permittivity of a medium and the dielectric constant

Permittivity of any material shares the bracket. Relative permittivity, also called the dielectric constant, is the ratio ε/ε₀, so it has no dimensions and no unit.

Permittivity and related quantities
Quantity	Relation	Dimensions
Permittivity of free space ε₀	From Coulomb's law	[M⁻¹L⁻³T⁴I²]
Permittivity of a medium ε	ε = εᵣε₀	[M⁻¹L⁻³T⁴I²]
Dielectric constant εᵣ	ε ÷ ε₀	dimensionless
Capacitance C	ε₀ × length	[M⁻¹L⁻²T⁴I²]

The last row is why the farad per metre works as a unit: capacitance divided by a length gives permittivity. See the capacitance page for C on its own.

Two checks with electric constants

Check: the speed of light c = 1/√(μ₀ε₀)

Permeability of free space: μ₀ is [MLT⁻²I⁻²]. Multiply by ε₀, [M⁻¹L⁻³T⁴I²].

M cancels, L goes to L¹⁻³ = L⁻², T to T⁻²⁺⁴ = T², I cancels. So μ₀ε₀ is [L⁻²T²].

One over its square root is [LT⁻¹], a speed. The relation passes, and it links ε₀ to the permeability of free space.

Check: parallel plate capacitance C = ε₀A/d

ε₀ times area over distance: [M⁻¹L⁻³T⁴I²] × [L²] ÷ [L] = [M⁻¹L⁻²T⁴I²].

That is the bracket of capacitance, so the formula is consistent.

Where marks are lost

Forgetting to square the charge. Two charges multiply, so current and time each appear squared before you flip the force.
Writing T⁻⁴ or T². The force in the denominator brings T⁻², which becomes +2 when inverted, added to the +2 from charge.
Giving εᵣ a unit. Relative permittivity is a pure ratio.
Using an old charge based system without saying so. Some older books treat charge Q as a base quantity and write [M⁻¹L⁻³T²Q²]. In SI, with current as base, the answer is [M⁻¹L⁻³T⁴I²].

Electric and magnetic constants are grouped with the other brackets on the dimensional formulas page.

Frequently asked questions

Why does mass have a negative power in ε₀?

Force sits in the denominator when ε₀ is made the subject of Coulomb's law. Force carries M¹, so ε₀ ends up with M⁻¹.

What is the dimensional formula of permittivity of a medium?

It equals that of ε₀, [M⁻¹ L⁻³ T⁴ I²], because the medium's permittivity is just a dimensionless number times ε₀.

In which SI unit is ε₀ given?

In C²/(N m²), or equivalently farad per metre, F/m. Its value is 8.854 × 10⁻¹² in either.

Does the dielectric constant have dimensions?

No. The dielectric constant is the ratio of two permittivities, so all powers cancel.

How are ε₀ and μ₀ connected?

Their product is one over the speed of light squared, μ₀ε₀ = 1/c². In dimensions, [L⁻²T²].