Speed of Light Formula: c = fλ, Value 3 × 10⁸ m/s and Examples

The speed of light formula used in most problems is $c = f\lambda$ : for any light wave or other electromagnetic wave in vacuum, frequency times wavelength equals c, which is 3.00 × 10⁸ m/s. A second formula explains where that number comes from, $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$ , built from the electric and magnetic constants of empty space.

Light is the fastest thing there is. The exact value in vacuum is 299 792 458 m/s, a number fixed by definition, and school problems round it to 3.00 × 10⁸ m/s. In a material such as water or glass light travels more slowly, at $v = \frac{c}{n}$ . This page shows each form, works examples from radio waves to sunlight, and lists the slips that cost marks in Class 12 optics and in AP Physics 2 tests.

Quick facts

Wave form
$c = f\lambda$
From constants
$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$
Value in vacuum
c = 3.00 × 10⁸ m/s (exactly 299 792 458 m/s)
In a medium
$v = \frac{c}{n}$
SI unit
metre per second (m/s)
Dimensional formula
[M⁰ L¹ T⁻¹]
Applies to
All electromagnetic waves: radio, microwaves, infrared, visible light, ultraviolet, X rays and gamma rays

Formula for speed of light, piece by piece

Symbols used for the speed of light
Symbol	Meaning	Value or unit
$c$	Light speed in empty space	about 3.00 × 10⁸ m/s
$f$	Frequency of the wave (ν in some books)	Hz
$\lambda$	Wavelength	m
$\mu_0$	Magnetic constant (permeability of vacuum)	4π × 10⁻⁷ T m/A
$\varepsilon_0$	Electric constant (permittivity of vacuum)	8.854 × 10⁻¹² C²/(N m²)
$n$	Refractive index of a medium	no unit

The wave form is the general wave velocity formula with v replaced by c. Since c is fixed in vacuum, a high frequency always pairs with a short wavelength. Gamma rays and radio waves differ only in f and λ; their product is the same.

Deriving c from μ₀ and ε₀

Electromagnetic theory predicts that a changing electric field makes a magnetic field and a changing magnetic field makes an electric field. The two can keep each other going and travel through empty space as a wave. The theory also gives the speed of that wave in terms of $\mu_0$ and $\varepsilon_0$ :

$c = \frac{1}{\sqrt{(4\pi \times 10^{-7})(8.854 \times 10^{-12})}} = \frac{1}{\sqrt{1.113 \times 10^{-17}}} = \frac{1}{3.336 \times 10^{-9}} \approx 3.00 \times 10^{8}\ \text{m/s}$

The match between this number and the measured speed of light was the evidence that light is an electromagnetic wave. Class 12 students meet this result in the electromagnetic waves chapter. The units also work out: you can show that $\frac{1}{\sqrt{\mu_0\varepsilon_0}}$ has units of m/s, which the dimensional formula of speed of light page does line by line.

Solved problems using the speed of light formula

Example 1: frequency of green light

Green light has a wavelength of 500 nm. Find its frequency.

Convert: λ = 500 nm = 5.00 × 10⁻⁷ m.

$f = \frac{c}{\lambda} = \frac{3.00 \times 10^{8}}{5.00 \times 10^{-7}} = 6.00 \times 10^{14}\ \text{Hz}$

Example 2: a microwave signal

A microwave link uses a frequency of 10 GHz. What is the wavelength?

Convert: f = 10 GHz = 1.0 × 10¹⁰ Hz.

$\lambda = \frac{c}{f} = \frac{3.00 \times 10^{8}}{1.0 \times 10^{10}} = 0.030\ \text{m} = 3.0\ \text{cm}$

Example 3: a laser pulse to the Moon

A laser pulse is fired at a reflector on the Moon, 3.84 × 10⁸ m away. How long until the echo returns?

$t = \frac{2d}{c} = \frac{2 \times 3.84 \times 10^{8}}{3.00 \times 10^{8}} = 2.56\ \text{s}$

The factor 2 is there because the pulse travels out and back.

Example 4: light inside water

Find the speed of light in water, n = 1.33.

$v = \frac{c}{n} = \frac{3.00 \times 10^{8}}{1.33} = 2.26 \times 10^{8}\ \text{m/s}$

Light year and other distance links

Because c is so large, astronomers measure distance by how far light goes in a given time. A light year is the distance light travels in one year of about 3.156 × 10⁷ s:

$1\ \text{ly} = c \times t = (3.00 \times 10^{8})(3.156 \times 10^{7}) \approx 9.46 \times 10^{15}\ \text{m}$

A light year is a distance, not a time, a point that catches out many students. The same idea in reverse lets radar and laser ranging measure distance: time a pulse there and back, multiply by c and halve it. The speed, distance and time formula is all you need.

Mistakes to watch for

Forgetting nanometres. 1 nm = 10⁻⁹ m. A wavelength of 600 nm is 6.00 × 10⁻⁷ m, not 6.00 × 10⁻⁵ m.
Using c in glass. Inside a material use v = c / n. Only in vacuum (and very nearly in air) is the speed c.
Changing frequency on refraction. Light keeps its frequency when it enters glass; its wavelength shrinks by the factor n. See the index of refraction formula.
Using the speed of sound. Sound needs a medium and moves about a million times more slowly. Compare the speed of sound formula.

In relativity c also sets a speed limit and links mass and energy through $E = mc^2$ , covered on the theory of relativity page.

Frequently asked questions

What is the formula of speed of light?

In vacuum c = fλ, frequency times wavelength. From electromagnetic theory c = 1/√(μ₀ε₀). Both give roughly 300 million metres every second.

What is the speed of light in km/s?

About 3.00 × 10⁵ km/s, or 300 000 km every second. Divide 3.00 × 10⁸ m/s by 1000 to convert.

Is the speed of light the same in all media?

No. It is greatest in vacuum. In a material it drops to c/n, so in glass with n = 1.5 it is about 2.0 × 10⁸ m/s.

Do all colours of light travel at the same speed?

In vacuum, yes: every colour and every electromagnetic wave moves at c. Inside glass, violet travels slightly slower than red, which causes dispersion.

What is the dimensional formula of the speed of light?

Like any speed it reduces to [M⁰ L¹ T⁻¹] in bracket notation, with m/s as the SI unit.