The source of electromagnetic waves can be a charge (a) moving with a constant velocity. (b) moving in a circu

NCERT Exemplar · Q31

Q. The source of electromagnetic waves can be a charge

(a) moving with a constant velocity.

(b) moving in a circular orbit.

(d) falling in an electric field.

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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 $v = f\lambda$ . …

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. …

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Electromagnetic waves are radiated only by an accelerating charge. A charge at rest or moving with constant velocity has unchanging fields and does not radiate. Of the four options, only (b) (moving in a circular orbit — continuous centripetal acceleration) and (d) (falling in an electric field — force produces acceleration) involve acceleration, so these are the correct choices; (a) and (c) are not.

Why acceleration is the key

A stationary charge has a static electric field around it — nothing changes with time, so there is no disturbance to propagate outward. A charge moving with constant velocity carries its field along with it, but in any fixed inertial frame the field pattern at a given point still only changes because the charge moves past it — there is no genuine "kink" that detaches and radiates away. It is only when the charge's velocity changes — that is, when it accelerates — that the field lines develop a travelling disturbance that propagates outward at speed $c$ as an electromagnetic wave.

P=\frac{q^2a^2}{6\pi\varepsilon_0 c^3}

Checking each option

Moving with a constant velocity. Constant velocity means zero acceleration ( $a=0$ ). By Larmor's formula, the radiated power is zero. Does not radiate — false.
Moving in a circular orbit. Even though the speed may be constant, the direction of the velocity is continuously changing, which means there is a non-zero (centripetal) acceleration at every instant, always directed toward the centre of the orbit. Radiates — true.
At rest. Zero velocity and, since nothing is causing it to move, zero acceleration. Does not radiate — false. …

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Q.The source of electromagnetic waves can be a charge

Concept understanding — Speed Of Light

Speed of Light

What It Is

Where the Value Comes From (Maxwell)

Key Properties

Speed of Light in a Medium

Why this formula?

Speed of Light: Why the Formula Holds

1. The Core Formula

2. Why This Formula? — The Derivation

Step 1: Maxwell's Equations

Step 2: The Wave Emerges

Step 3: The Wave Equation

3. Why Is It Constant?

Why acceleration is the key

Checking each option

Method: Deciding Whether a Given Charge Motion Radiates Electromagnetic Waves

Steps

Q.The source of electromagnetic waves can be a charge

Why acceleration is the key

Checking each option

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