Q.In a plane electromagnetic wave travelling along the $x$-axis, the electric field oscillates along the $y$-axis. State along which axis the magnetic field of this wave must oscillate, and explain how this direction is fixed by the direction of propagation.

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).

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).

A sketch of this would show, along a horizontal x-axis: a sinusoidal curve in the x-y plane representing E (oscillating up-down in y), and simultaneously a sinusoidal curve in the x-z plane representing B (oscillating in-out in z), both sine curves in phase with each other, both perpendicular to the x-axis and to each other, propagating together along +x.

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.

Step 2 — direction: The wave travels along +X (î) and B is along +Z (k̂). E, B and the propagation vector obey E × B ∝ direction of propagation. Since ĵ × k̂ = î, E must point along +Y (ĵ).

Step 3 — combine: E = 7·2 ĵ V/m.

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).

Why $0^\circ$. Both planes are drawn through the line of propagation. Since the direction of propagation actually lies in the plane of polarization, the angle a line makes with a plane that contains it is $0^\circ$. (The two planes — vibration and polarization — are mutually perpendicular, i.e. $90^\circ$ to each other, but that is a different quantity from what is asked.)

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 classified as transverse waves — this is confirmed experimentally by the fact that light (an EM wave) can be polarised, a property unique to transverse waves.

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:

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

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

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

4. 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.

5. They carry energy and momentum, and exert radiation pressure.

6. They are produced by accelerating (oscillating) electric charges.

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 $z$:

$\hat{y}\times\hat{z} = \hat{x}$

which is exactly the direction of propagation. Neither $\vec E$ alone nor $\vec B$ alone can indicate the propagation direction (they lie transverse to it), and $\vec E\cdot\vec B = 0$ always (they are perpendicular), so it carries no directional information. Only the vector (cross) product $\vec E\times\vec B$ correctly points along the direction of travel.

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 set). They reach their maximum and minimum values at the same place and instant — i.e. they are in phase.

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 $\vec{k}$:

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

Because the oscillating quantities are at right angles to the propagation direction, EM waves are transverse. This is confirmed by their ability to be polarised (a property only transverse waves possess). They do not need a material medium, unlike longitudinal sound waves.

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:

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

So the wave travels parallel to E × B, not parallel to E or B individually, and not to B × E (opposite direction).

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

In an EM wave, $\vec E$ and $\vec B$ oscillate perpendicular to each other and both are perpendicular to the direction of propagation, forming a right-handed set $(\vec E, \vec B, \vec v)$. The direction of propagation is therefore along $\vec E \times \vec B$ (this also matches the direction of the Poynting vector $\vec S = \frac{1}{\mu_0}\vec E \times \vec B$, which gives the direction of energy flow).

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 $\vec{k}$). Since the oscillating quantities (E and B) are perpendicular to the direction of travel, electromagnetic waves are classified as transverse waves — unlike sound, which is longitudinal.