(a) On what factors does the speed of an electromagnetic wave in a medium depend ? (b) How is an electromagnet

Question

Q. (a) On what factors does the speed of an electromagnetic wave in a medium depend ?

(b) How is an electromagnetic wave produced ?

(c) Sketch a schematic diagram depicting the electric and magnetic fields for an electromagnetic wave propagating along z-axis.

CBSECBSE Class XII Board 2024Subjective· 3mImportance★★★★★

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The speed of an EM wave in a medium depends on the medium’s electric permittivity and magnetic permeability. An EM wave is produced by an accelerating charge. For propagation along the z-axis, the electric field oscillates along x and the magnetic field along y, both in phase.

Schematic of an electromagnetic wave propagating along the z-axis: the electric field E oscillates as a sine wave along the x-axis and the magnetic field B oscillates as a sine wave along the y-axis, the two mutually perpendicular fields in phase with each other, with the wave travelling in the direction given by E x B, i.e. along z.

The Concept and Intuition

Electromagnetic waves are the self-sustaining dance of electric and magnetic fields. When a charged particle accelerates, it disturbs the electric field around it. This changing electric field, by Maxwell’s laws, creates a magnetic field. A changing magnetic field, in turn, creates an electric field. These two fields regenerate each other, allowing the disturbance to travel outward as a wave — even through empty space.

The speed of this wave is not arbitrary. It is determined by how easily the medium allows electric and magnetic fields to form. The electric permittivity ( $\varepsilon$ ) tells us how much the medium resists the formation of an electric field; the magnetic permeability ( $\mu$ ) tells us the same for magnetic fields. The wave’s speed is the inverse square root of their product.

Step-by-Step Solution

1. Factors affecting the speed of an EM wave in a medium

The speed $v$ of an electromagnetic wave in a linear, isotropic, non-magnetic medium is given by:

v = \frac{1}{\sqrt{\mu \varepsilon}}

where $\mu$ is the magnetic permeability of the medium and $\varepsilon$ is its electric permittivity.

For a vacuum, these take their minimum values: $\mu_0$ and $\varepsilon_0$ , giving the universal constant $c = 1/\sqrt{\mu_0 \varepsilon_0} \approx 3 \times 10^8$ m/s.

In any material medium, both $\mu$ and $\varepsilon$ are larger than their vacuum values. This means the wave always slows down compared to vacuum. The ratio $c/v = \sqrt{\mu_r \varepsilon_r}$ is called the refractive index of the medium, where $\mu_r = \mu/\mu_0$ and $\varepsilon_r = \varepsilon/\varepsilon_0$ are the relative permeability and permittivity.

Watch out

A common mistake is to think the speed depends on frequency or amplitude. It does not — for a given medium, $\mu$ and $\varepsilon$ are (ideally) constants. However, in dispersive media, $\varepsilon$ can vary with frequency, causing different colours (frequencies) to travel at slightly different speeds — that is how a prism splits white light.

2. How an electromagnetic wave is produced

An electromagnetic wave is produced whenever an electric charge accelerates. This includes:

A charge oscillating back and forth (like in a radio antenna)
A charge moving in a circle (like an electron in a synchrotron)
Any change in the velocity of a charged particle

The reason is rooted in Maxwell’s equations. A changing current (which is moving charge) produces a changing magnetic field. That changing magnetic field induces a changing electric field, and so on. The disturbance propagates outward at speed $v$ , carrying energy away from the source. …

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CBSE 2026Set 55/2/11 markMCQ
$x$
›Reveal solutionSolution
In a propagating electromagnetic wave, the electric and magnetic fields oscillate in phase with each other, so the phase difference is zero. The correct option is (A).

Concept and Intuition

The question tests a fundamental property of electromagnetic waves in free space. Many students carry a vague memory that $\vec{E}$ and $\vec{B}$ are "perpendicular" and mistakenly think that means a $\pi/2$ phase difference. But perpendicularity here refers to direction in space, not a time delay.

In a plane electromagnetic wave, both fields vary sinusoidally with position and time. Maxwell's equations demand that the time-varying electric field produces the magnetic field, and vice versa — they are coupled in such a way that their peaks and zeros occur at the same instant. There is no lag between them.

Watch out

A common mistake is to confuse spatial orthogonality (the fields are perpendicular to each other and to the direction of propagation) with a phase difference. They are perpendicular in space, but they oscillate in time together — zero phase difference.

Step-by-Step Reasoning

Recall the wave equations from Maxwell's laws In free space (no charges or currents), Maxwell's equations yield wave equations for both $\vec{E}$ and $\vec{B}$ . For a wave traveling along the $x$ -axis, the solutions are:

$E_y = E_0 \sin(\omega t - kx + \phi)$ $B_z = B_0 \sin(\omega t - kx + \phi)$
Notice the same argument $(\omega t - kx + \phi)$ appears in both. This is not an assumption — it follows directly from Faraday's law and Ampère's law.

Why the phase must be identical Consider Faraday's law for a plane wave:

$\frac{\partial E_y}{\partial x} = -\frac{\partial B_z}{\partial t}$
If $E_y = E_0 \sin(\omega t - kx + \phi_E)$ and $B_z = B_0 \sin(\omega t - kx + \phi_B)$ , then:

Left side: $\frac{\partial E_y}{\partial x} = -k E_0 \cos(\omega t - kx + \phi_E)$

Right side: $-\frac{\partial B_z}{\partial t} = -\omega B_0 \cos(\omega t - kx + \phi_B)$ …
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CBSE 2026Set ANNUAL1 mark
$Q. Draw propagation diagram of a linearly polarised electromagnetic wave.$
›Reveal solutionSolution
Figure — This is a 1-mark 'Draw propagation diagram of a linearly polarised EM wave' item where the drawing IS the answ

In a linearly polarised electromagnetic wave, the electric and magnetic fields oscillate along two fixed, mutually perpendicular directions, both perpendicular to the direction the wave travels.

Consider a wave travelling along the x-direction. In a linearly (plane) polarised wave, the electric field vector E oscillates only along one fixed direction, say the y-axis: Ey = E0 sin(kx - omega t). The magnetic field vector B oscillates only along the z-axis, in phase with E: Bz = B0 sin(kx - omega t), with E0/B0 = c (speed of light). …

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CBSE 2026Set SEM31 markMCQ
$Q. The electromagnetic wave travels in free space along X-direction. At a particular point in space and time, B is given by 2\cdot4 \times 10⁻⁸ k̂ (in Tesla unit), then at that point E in V/m unit is$
(a) 7·2 ĵ
(b) 7·2 î
(c) 0·8 k̂
(d) 2·4 k̂
›Reveal solutionSolution

In free space E = cB in magnitude, and E, B, and the propagation direction form a right-handed set. Magnitude = 7·2 V/m; direction = ĵ. Option (a).

Step 1 — magnitude: For an electromagnetic wave in vacuum, E = cB = (3×10⁸ m/s)(2·4×10⁻⁸ T) = 7·2 V/m.

…

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$0^\circ$
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The propagation direction lies inside the plane of polarization, so the angle between the plane of polarization and the direction of propagation is $0^\circ$ — option (A).

Concept. An electromagnetic wave is transverse: the electric field $\vec E$ and magnetic field $\vec B$ oscillate perpendicular to the direction of propagation. Two planes are defined:

Plane of vibration — contains the electric-field vibration and the direction of propagation.

Plane of polarization — passes through the direction of propagation and is perpendicular to the plane of vibration (it carries no vibrations). …
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Electromagnetic waves are transverse in nature.

In an electromagnetic wave, the electric field vector E and the magnetic field vector B oscillate sinusoidally, perpendicular to each other, and both are perpendicular to the direction in which the wave travels. Since the oscillating quantities (E and B) vary in a direction perpendicular to the direction of propagation, electromagnetic waves are classif …

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CBSE 2025Set ANNUAL1 mark
$Q. What is the nature of electromagnetic waves?$
›Reveal solutionSolution
EM waves consist of time-varying electric and magnetic fields, perpendicular to each other and to the direction the wave travels, and unlike sound they can propagate through vacuum.

Key features of the nature of electromagnetic waves:

They are transverse waves: the oscillating E and B fields are perpendicular to the direction of wave propagation.

E and B are mutually perpendicular to each other as well, and both oscillate in phase.

They require no material medium and can travel through vacuum, unlike mechanical waves (e.g. sound).

In vacuum they all travel at the same speed, c = 3 x 10^8 m/s, related to the fields' amplitudes by c = E0/B0. …
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CBSE 2024Set ANNUAL1 markMCQ
$\vec{E}$
(a) $\vec{E}$
(b) $\vec{B}$
(c) $\vec{E}\times\vec{B}$
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›Reveal solutionSolution

In an electromagnetic wave, $\vec E$ and $\vec B$ are mutually perpendicular and both perpendicular to the direction the wave travels; the direction of propagation is given by the cross product $\vec E \times \vec B$ (the direction of the Poynting vector $\vec S = \frac{1}{\mu_0}\vec E\times\vec B$ ).

Reasoning

For a plane e.m. wave travelling along, say, the $x$ -axis, with $\vec E$ along $y$ and $\vec B$ along $z$ :

$\hat{y}\times\hat{z} = \hat{x}$ …

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›Reveal solutionSolution

In an EM wave, E and B oscillate perpendicular to each other and to the direction of propagation, and they are in phase.

Maxwell's equations show that an electromagnetic wave consists of oscillating electric field $\vec{E}$ and magnetic field $\vec{B}$ , both perpendicular to the direction of wave propagation and to each other (i.e. $\vec E, \vec B$ and the propagation direction $\vec k$ form a mutually perpendicular …

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CBSE 2022Set I1 markMCQ
$Q. Nature of electromagnetic waves is (A) transverse (B) longitudinal (C) both (A) and (B) (D) electrical$
›Reveal solutionSolution

EM waves are transverse.

In an electromagnetic wave the electric field $\vec{E}$ and magnetic field $\vec{B}$ oscillate perpendicular to each other and both are perpendicular to the direction of propagation $\vec{k}$ :

$\vec{E} \perp \vec{B} \perp \vec{k}.$

…

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CBSE 2021Set A1 markMCQ
$Q. The direction of propagation of electromagnetic wave is (A) parallel to E (B) parallel to B (C) parallel to (B \times E) (D) parallel to (E \times B)$
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An EM wave propagates in the direction of E × B.

In an electromagnetic wave the electric field E, magnetic field B, and the direction of propagation are mutually perpendicular, forming a right-handed set. The direction of energy flow (Poynting vector S = (1/μ₀) E × B) gives the direction of propagation:

$\hat{k} \parallel (\vec{E} \times \vec{B})$ …

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An electromagnetic wave travels in the direction given by the cross product of its electric and magnetic field vectors.

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Electromagnetic waves are transverse waves — E and B oscillate at right angles to the direction the wave travels.

In an electromagnetic wave, the electric field vector $\vec{E}$ and the magnetic field vector $\vec{B}$ oscillate sinusoidally, and both are always perpendicular to each other AND perpendicular to the direction of propagation of the wave $\vec{k}$ (i.e. $\vec{E}, \vec{B}, \vec{k}$ form a mutually perpendicular set, with $\vec{E}\times\vec{B}$ along $\vec{k}$ ). S …

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Q.(a) On what factors does the speed of an electromagnetic wave in a medium depend ?

Concept understanding — Transverse Nature of Electromagnetic Waves

The Concept and Intuition

Step-by-Step Solution

Concept and Intuition

Step-by-Step Reasoning

Q.(a) On what factors does the speed of an electromagnetic wave in a medium depend ?

The Concept and Intuition

Step-by-Step Solution

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