Because c is simply a speed, the speed of light dimensional formula is [M⁰ L¹ T⁻¹], short form [LT⁻¹]. Its value in vacuum is 3.00 × 10⁸ m/s, but the bracket is the same as for a cyclist or a sound wave.
What makes c special is not its dimensions but the way it ties other constants together. It is distance over time, it is frequency times wavelength, and it is one over the square root of μ₀ε₀. All three must give [LT⁻¹], and they do. This page works through each route, the unit, a light year trap, a mass energy check and the slips. More on the value of c and light travel times sits with the speed of light formula.
Quick facts
Dimensional formula
[M⁰ L¹ T⁻¹], short form [LT⁻¹]Value in vacuum
3.00 × 10⁸ m/sRelations
,SI unit
m/sScalar or vector
Speed c is a scalarChapter
Electromagnetic waves, Class 12
Three derivations of the speed of light dimensional formula
As a distance over a time
c = d/t. A distance is [L] and a time is [T], so c is [LT⁻¹].
As frequency times wavelength
c = fλ. Frequency is [T⁻¹] and wavelength is [L]. The product is [LT⁻¹] again.
From the electric and magnetic constants
c = 1/√(μ₀ε₀). With μ₀ as [MLT⁻²I⁻²] and ε₀ as [M⁻¹L⁻³T⁴I²], the product μ₀ε₀ loses its M and I and becomes [L⁻²T²]. Taking the inverse square root turns that into [LT⁻¹].
The third route is the deepest: it shows that light is an electromagnetic wave whose speed is fixed by two constants of electricity and magnetism. The permittivity of free space page has the ε₀ bracket.
Unit, exact value and conversions
[LT⁻¹] becomes m s⁻¹ in base units, written m/s. Since the metre is now defined from the speed of light, c is exactly 299 792 458 m/s; school problems use 3.00 × 10⁸ m/s.
Express c in km/h and in cm/s
km/h: multiply by 3600 s/h and divide by 1000 m/km, a net factor of 3.6. So c = 3.00 × 10⁸ × 3.6 = 1.08 × 10⁹ km/h.
cm/s: a metre is 100 cm and length appears once, so c = 3.00 × 10¹⁰ cm/s.
Find the travel time of sunlight over the 1.50 × 10¹¹ m to Earth
t = d/c = 1.50 × 10¹¹ ÷ 3.00 × 10⁸ = 500 s, about 8.3 minutes.
The answer has units of m ÷ (m/s) = s, as dimensions predict.
Other speeds and two look alikes
| Quantity | Typical value | Note |
|---|---|---|
| c in a vacuum | 3.00 × 10⁸ m/s | A universal constant |
| Speed of light in glass | about 2 × 10⁸ m/s | c divided by refractive index |
| Speed of sound in air | about 340 m/s | A mechanical wave |
| Speed of a satellite in low orbit | about 7.8 × 10³ m/s | Orbital speed |
Two quantities near c are often confused with it. The refractive index n = c/v divides one speed by another, so it has no dimensions. A light year measures how far light gets in one year, so it is a length, [L], not a speed or a time. See the velocity page for speed in general.
E = mc² as a dimension check
Check: mass energy relation E = mc²
Right side: [M] × [LT⁻¹]² = [ML²T⁻²].
That is the energy bracket, so E = mc² is dimensionally sound. It also shows that c² converts a mass into an energy: one kilogram corresponds to 9.0 × 10¹⁶ J.
A quick test for any relativity formula: the factor v/c or v²/c² must appear wherever a speed is compared with c, because only a ratio of speeds is dimensionless. The energy page lists the bracket.
Errors to watch for
- Calling the light year a unit of time. It is a distance, about 9.46 × 10¹⁵ m.
- Giving refractive index the unit m/s. n is a ratio of speeds and has no unit.
- Treating c as dimensionless because it is a constant. Being constant does not remove dimensions; c is [LT⁻¹].
- Getting [L⁻¹T] from 1/√(μ₀ε₀). The root of [L⁻²T²] is [L⁻¹T]; the "one over" then flips it to [LT⁻¹].
Physical constants appear with their brackets in the dimensional formulas collection.
Frequently asked questions
Is the speed of light a scalar?
The speed c is a scalar. The velocity of a light beam is a vector with size c and the direction of travel.
Which SI unit does c take?
Metre per second, m/s. Its exact value is 299 792 458 m/s, usually rounded to 3.00 × 10⁸ m/s.
Does the speed of light have the same dimensions as the speed of sound?
Yes. Any speed, however large or small, reduces to [LT⁻¹].
Is a light year a speed?
No. A light year is a distance, the length light covers in one year, so its dimensions are [L].
How does c relate to μ₀ and ε₀?
c = 1/√(μ₀ε₀). The product μ₀ε₀ has dimensions [L⁻²T²], so its inverse square root is a speed.