Q.The speed of electromagnetic waves in vacuum is given by the equation
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Speed of Light
What It Is
The speed of light is the speed at which light — and every other electromagnetic wave — travels through empty space (vacuum). It is one of the most important constants in physics, denoted c:
c≈3×108 m/s=3×105 km/s
More precisely c=2.998×108 m/s. In one second light travels about 300,000 km — roughly seven and a half times around the Earth.
Where the Value Comes From (Maxwell)
The speed of light is not just measured; it is predicted by Maxwell's equations. When Maxwell combined his laws of electricity and magnetism, he found that electromagnetic waves must travel through vacuum at a speed fixed entirely by two constants of free space:
c=μ0ε01
where
- ε0=8.85×10−12 C2N−1m−2 is the permittivity of free space, and
- μ0=4π×10−7 T m A−1 is the permeability of free space.
Plugging in these numbers gives c≈3×108 m/s — matching the measured speed of light. This agreement was the decisive clue that light itself is an electromagnetic wave.
Key Properties
- Same for all electromagnetic waves. Radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays all travel at c in vacuum, regardless of their frequency or wavelength.
- Universal constant. In vacuum, c is the same for every observer and does not depend on the motion of the source — the starting postulate of Einstein's special relativity.
- The cosmic speed limit. No material object or signal carrying information can travel faster than c.
- Links wavelength and frequency. For any EM wave in vacuum,
c=fλ
so a high-frequency wave has a short wavelength and vice versa.
Speed of Light in a Medium
Inside a transparent material (glass, water, etc.) light slows down. Its speed becomes
v=nc=με1
where n=μrεr is the refractive index of the medium and is always greater than 1. For example, in water n≈1.33, so light travels at about 2.25×108 m/s. The frequency stays the same, but the wavelength shortens because v=fλ. …
Why this formula?
Speed of Light: Why the Formula Holds
The speed of light (c) is not just a number — it emerges from the fundamental laws of electricity and magnetism. Let's understand why its value is fixed and where the formula comes from.
1. The Core Formula
The speed of light in vacuum is given by:
c=μ0ε01
Where:
- μ0 = permeability of free space (how easily a magnetic field forms)
- ε0 = permittivity of free space (how easily an electric field forms)
2. Why This Formula? — The Derivation
Step 1: Maxwell's Equations
James Clerk Maxwell unified electricity and magnetism into four equations. Two key ones for light:
- Faraday's Law: A changing magnetic field creates an electric field
∇×E=−∂t∂B
- Ampère's Law (with Maxwell's correction): A changing electric field creates a magnetic field
∇×B=μ0ε0∂t∂E
Step 2: The Wave Emerges
Take the curl of Faraday's Law:
∇×(∇×E)=−∂t∂(∇×B)
Using the vector identity ∇×(∇×E)=∇(∇⋅E)−∇2E and noting that in vacuum ∇⋅E=0, we get:
−∇2E=−∂t∂(∇×B)
Now substitute Ampère's Law for ∇×B:
−∇2E=−∂t∂(μ0ε0∂t∂E)
Step 3: The Wave Equation
This simplifies to:
∇2E=μ0ε0∂t2∂2E
This is the wave equation. For any wave, the general form is:
∇2E=v21∂t2∂2E
Comparing the two, the wave speed v must satisfy:
v21=μ0ε0
Hence:
v=μ0ε01
This v is the speed of light — denoted c.
3. Why Is It Constant?
- μ0 and ε0 are fundamental constants of nature — they don't depend on the observer or the source.
- Therefore, c is also a universal constant. …
Maxwell derived the speed of an electromagnetic wave in vacuum directly from his equations for the electric and magnetic fields, and the value that fell out matched the already-known speed of light. …
Maxwell showed that combining the wave equations for E and B derived from his equations in free space gives a wave speed c = 1/sqrt(mu0 epsilon0), which matched the known speed of light and led him to conclude light itself is an electromagnetic wave.
From Maxwell's equations in vacuum, the electric and magnetic fields each satisfy a wave equation with speed:
c = 1 / sqrt(mu0 epsilon0)
…
- CBSE 2026Set ANNUAL1 markMCQQ.Assertion (A): The speed of electromagnetic waves in vacuum is same for all wavelengths. Reason (R): Formula for the speed of electromagnetic waves in vacuum is c=μ0ϵ01.(i) Both A and R are correct and R is correct explanation of A.(ii) Both A and R are correct but R is not the correct explanation of A.(iii) A is correct but R is incorrect.(iv) Both A and R are incorrect.
›Reveal solutionSolution
Both statements are true, and the formula in R is exactly why speed is wavelength-independent.
Assertion is correct: in vacuum every electromagnetic wave, regardless of wavelength, travels at the same speed c (vacuum is non-dispersive for EM waves). Reason is also correct: c=μ0ϵ01, and since μ0 and ϵ0 are constants of free space (not functions of wavelength or frequency), t …
- CBSE 2025Set 55/5/11 markMCQQ.The dimensions of (με)−1, where ε is the permittivity and μ the permeability of a medium, are: (A) [M0L1T−1] (B) [M0L2T−2] (C) [M1L2T−2] (D) [M1L−1T1]
›Reveal solutionSolution
The product με has dimensions of [T2L−2], so (με)−1 has dimensions of velocity squared: [L2T−2]. The answer is (B).
The key insight here is recognizing what μ and ε represent physically and how they combine in electromagnetic theory. The speed of light (or electromagnetic waves) in a medium is given by c=με1, which immediately tells us that με must have dimensions of inverse velocity squared.
Let me work through the dimensional analysis systematically.
Finding the dimensions of permittivity ε:
Permittivity appears in Coulomb's law and the relation for electric field. From the capacitance of a parallel-plate capacitor:
C=dεA
where C is capacitance, A is area, and d is separation. Since capacitance has dimensions [Q2T2M−1L−2] (from C=Q/V and energy U=21CV2), we get:
[ε]=[L2][C][L]=[M−1L−3T4A2]
In SI base units with charge Q=AT, this becomes [ε]=[M−1L−3T4A2].
Finding the dimensions of permeability μ:
Permeability appears in Ampère's law. The magnetic field around a wire is B=2πrμI, giving:
[μ]=[A][B][L] …
- CBSE 2025Set ANNUAL1 markMCQQ.The speed of electromagnetic waves in vacuum is given by the equation(a) c = sqrt(mu0 * epsilon0)(b) c = 1 / sqrt(mu0 * epsilon0)(c) c = sqrt(mu0 / epsilon0)(d) c = sqrt(epsilon0 / mu0)
›Reveal solutionSolution
Maxwell showed that combining the wave equations for E and B derived from his equations in free space gives a wave speed c = 1/sqrt(mu0 epsilon0), which matched the known speed of light and led him to conclude light itself is an electromagnetic wave.
From Maxwell's equations in vacuum, the electric and magnetic fields each satisfy a wave equation with speed:
c = 1 / sqrt(mu0 epsilon0)
…
- CBSE 2025Set ANNUAL1 markMCQQ.The dimension of μ0ε01 is :(a) [L−1T](b) [LT−1](c) [L−2T2](d) [L2T−2]
›Reveal solutionSolution
Since 1/μ0ε0=c (the speed of light), 1/(μ0ε0)=c2, whose dimension is [L2T−2].
Working
Maxwell's relation gives the speed of electromagnetic waves in vacuum as
c=μ0ε01
Squaring both sides:
c2=μ0ε01
…
- CBSE 2023Set F1 markMCQQ.Light year is equal to (A) 9.46 x 10^15 m (B) 9.46 x 10^12 m (C) 9.46 x 10^8 m (D) 9.46 x 10^10 m
›Reveal solutionSolution
1 light year = c × (1 year) ≈ 3×10⁸ m/s × 3.15×10⁷ s ≈ 9.46 × 10¹⁵ m.
A light year is the distance travelled by light (an electromagnetic wave) in vacuum in one year.
…
- CBSE 2023Set ANNUAL1 markQ.The velocity of electromagnetic waves in free space can be given by the relation ............. .
›Reveal solutionSolution
Maxwell's equations predict that electromagnetic waves travel in free space at a speed fixed by the permeability and permittivity of free space.
From Maxwell's electromagnetic wave equation, the speed of an electromagnetic wave in free space (vacuum) is
c=μ0ε01 …
- CBSE 2023Set ANNUAL1 markMCQQ.When medium of electromagnetic waves changes from air to water, their speed -(a) increases(b) decreases(c) remains same(d) may increase or decrease
›Reveal solutionSolution
Speed of light in a medium is v=c/n; water has n>1, so speed decreases relative to air.
The speed of an electromagnetic wave in a medium is v=nc, where n is the refractive index of that medium. Water has a refractive index greater than that of air (nwater≈1.33>nair≈1), so the speed of th …
- CBSE 2022Set ANNUAL1 markMCQQ.Relation between velocity of light (c), permeability of free space (μ₀), permittivity of free space (ε₀) is ____.(a) C = 1/(μ₀ε₀)(b) C = 1/√(μ₀ε₀)(c) C = μ₀ε₀(d) C = √(μ₀ε₀)
›Reveal solutionSolution
Maxwell showed that the speed of light equals 1/μ0ε0, one of the great unifications of physics.
…
- CBSE 2021Set A1 markMCQQ.The unit of which physical quantity is light year? (A) Distance (B) Time (C) Energy (D) Intensity of light
›Reveal solutionSolution
A light year is a unit of distance, not time.
Despite the word 'year', a light year is the distance travelled by light (speed c ≈ 3 × 10⁸ m/s) in one year:
1 light year=c×(1 year)≈9.46×1015 m
…
- CBSE 2021Set ANNUAL1 markMCQQ.The velocity of electromagnetic waves in free space is given by –(a) √(ε₀μ₀)(b) 1/√(ε₀μ₀)(c) ε₀/μ₀(d) √(ε₀/μ₀)
›Reveal solutionSolution
The speed of an EM wave in free space comes directly from Maxwell's equations and equals c=1/ε0μ0.
Maxwell's equations predict electromagnetic waves travelling in vacuum with speed:
v=ε0μ01 …
- CBSE 2019Set ANNUAL1 markMCQQ.If Vg, Vx and Vm are the speeds of gamma rays, X-rays and microwaves respectively in vacuum, then(a) Vg = Vx = Vm(b) Vg > Vx > Vm(c) Vg > Vx < Vm(d) Vg < Vx < Vm
›Reveal solutionSolution
Gamma rays, X-rays and microwaves are all part of the same electromagnetic spectrum, and every electromagnetic wave travels at the speed of light c in vacuum, independent of its frequency.
Gamma rays, X-rays, and microwaves differ enormously in frequency and wavelength (gamma rays have the shortest wavelength/highest frequency, microwaves the longest wavelength/lowest frequency among these three), but they are all transverse electromagnetic waves consisting of oscillating electric and magnetic fields. A defining property of electromagnetic waves derived from Maxwell's equations is that in vacuum they all propagate at the same universal speed, …
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