The dimension of $\dfrac{1}{\mu_0\epsilon_0}$ is (a) $[LT^{-1}]$ (b) $[L^2T^{-2}]$ (c) $[L^{-1}T]$ (d) $[L^{-2}T^2]$

The dimension of (1)/(μ₀epsilon₀) is (a) [LT⁻¹] (b) [L²T⁻²] (c) [L⁻¹T] (d) [L⁻²T²]

I. Multiple Choice Questions · Q1

\dfrac{1}{\mu_0\epsilon_0}

(a)

[LT^{-1}]

(b)

[L^2T^{-2}]

(c)

[L^{-1}T]

(d)

[L^{-2}T^2]

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Concept understanding — Speed Of Light

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 \approx 3 \times 10^8 \ \text{m/s} = 3\times10^{5}\ \text{km/s}

More precisely $c = 2.998 \times 10^8\ \text{m/s}$ . 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 = \frac{1}{\sqrt{\mu_0\,\varepsilon_0}}

where

$\varepsilon_0 = 8.85\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}$ is the permittivity of free space, and
$\mu_0 = 4\pi\times10^{-7}\ \text{T m A}^{-1}$ is the permeability of free space.

Plugging in these numbers gives $c \approx 3\times10^8$ 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 = f\lambda

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 = \frac{c}{n} = \frac{1}{\sqrt{\mu\,\varepsilon}}

where $n = \sqrt{\mu_r\varepsilon_r}$ is the refractive index of the medium and is always greater than 1. For example, in water $n \approx 1.33$ , so light travels at about $2.25\times10^8$ m/s. The frequency stays the same, but the wavelength shortens because $v = f\lambda$ .

Watch out

When light passes from vacuum into a medium, do NOT change its frequency — only its speed and wavelength change. Frequency is set by the source.

A Quick Example

How long does sunlight take to reach the Earth, a distance of about $1.5\times10^{11}$ m?

t = \frac{d}{c} = \frac{1.5\times10^{11}}{3\times10^8} = 500\ \text{s} \approx 8.3\ \text{minutes}

The Big Picture

The speed of light ties electricity, magnetism and optics into a single framework. Because $c = 1/\sqrt{\mu_0\varepsilon_0}$ depends only on properties of empty space, it is a genuine constant of nature — and its constancy underpins both Maxwell's electromagnetism and Einstein's relativity.

Remember: $c = 3\times10^8$ m/s in vacuum, $c = 1/\sqrt{\mu_0\varepsilon_0}$ , same for all EM waves, and it slows to $v=c/n$ inside a medium.

The speed of light and its derivation from Maxwell's equations, c = 1/√(μ₀ε₀), is covered in the NCERT Class 12 Physics chapter on electromagnetic waves, tested regularly in CBSE boards, JEE Main and NEET. Students searching "speed of light formula and electromagnetic spectrum class 12 physics" will find this permittivity-permeability derivation matches exactly how NCERT presents the topic.

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 = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

Where:

$\mu_0$ = permeability of free space (how easily a magnetic field forms)
$\varepsilon_0$ = 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

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}

Ampère's Law (with Maxwell's correction): A changing electric field creates a magnetic field

\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}

Step 2: The Wave Emerges

Take the curl of Faraday's Law:

\nabla \times (\nabla \times \mathbf{E}) = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B})

Using the vector identity $\nabla \times (\nabla \times \mathbf{E}) = \nabla(\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E}$ and noting that in vacuum $\nabla \cdot \mathbf{E} = 0$ , we get:

-\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B})

Now substitute Ampère's Law for $\nabla \times \mathbf{B}$ :

-\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} \left( \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)

Step 3: The Wave Equation

This simplifies to:

\nabla^2 \mathbf{E} = \mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2}

This is the wave equation. For any wave, the general form is:

\nabla^2 \mathbf{E} = \frac{1}{v^2} \frac{\partial^2 \mathbf{E}}{\partial t^2}

Comparing the two, the wave speed $v$ must satisfy:

\frac{1}{v^2} = \mu_0 \varepsilon_0

Hence:

v = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}

This $v$ is the speed of light — denoted $c$ .

3. Why Is It Constant?

$\mu_0$ and $\varepsilon_0$ are fundamental constants of nature — they don't depend on the observer or the source.
Therefore, $c$ is also a universal constant.
This was a revolutionary idea: light doesn't need a medium (like "ether") — it's a self-propagating electromagnetic wave.

4. Key Insight for Exams

Concept	Why It Matters
$c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$	Shows light is an electromagnetic wave
No medium needed	Electric and magnetic fields sustain each other
$c$ is maximum speed	From special relativity — nothing with mass can reach it

5. Quick Numerical Check

\mu_0 = 4\pi \times 10^{-7} \, \text{H/m}

\varepsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}

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

This matches experiment perfectly — confirming Maxwell's theory.

Bottom line: The speed of light formula isn't arbitrary — it's a direct consequence of how electric and magnetic fields interact. The constants $\mu_0$ and $\varepsilon_0$ determine how fast this interaction propagates through space.

Step 1. The speed of an electromagnetic wave in free space is $c=\dfrac{1}{\sqrt{\mu_0\epsilon_0}}$ .

Step 2. Squaring both sides gives $c^2=\dfrac{1}{\mu_0\epsilon_0}$ , so the quantity $\dfrac{1}{\mu_0\epsilon_0}$ is numerically and dimensionally just $c^2$ , the square of a speed.

Step 3. Speed has dimension $[LT^{-1}]$ , so speed-squared has dimension $[LT^{-1}]^2=[L^2T^{-2}]$ .

Step 4. Eliminating the others: (a) $[LT^{-1}]$ is the dimension of $c$ itself, not $c^2$ ; (c) and (d) do not correspond to any speed-related quantity here.

✓Final answer

[L^2T^{-2}]

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