The E -field of an electromagnetic wave is described by 𝐄=(𝐢̂+𝐣̂) E0 sin(k z-ωt+π/ 6) Write an e

step 1 Hello students in this question we have given the electric field vector E vector it is equals to

The E -field of an electromagnetic wave is described by...

The $E$ -field of an electromagnetic wave is described by $$\overrightarrow{\mathbf{E}}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}) E_{0} \sin (k z-\omega t+\pi / 6)$$ Write an expression for the $B$ -field. Determine $\overrightarrow{\mathbf{B}}(0,0)$

Key Concepts

Plane Electromagnetic Waves

Plane electromagnetic waves are solutions to Maxwell’s equations in free space wherein the fields vary sinusoidally and propagate in a uniform direction. These waves have constant amplitude surfaces (planes) perpendicular to the direction of propagation. Their mathematical description often involves sinusoidal functions of space and time, which encapsulate the wave’s frequency, wavelength, and phase.

Transverse Nature of Electromagnetic Waves

In free space, electromagnetic waves are transverse, meaning the oscillating electric (E) and magnetic (B) fields are perpendicular to each other and to the direction of propagation. This orthogonality is a fundamental aspect of wave propagation according to Maxwell’s equations and ensures that energy is transmitted efficiently in the wave.

Relationship between E and B Fields in Free Space

The magnitudes of the electric and magnetic fields in an electromagnetic wave are intrinsically related by the speed of light (c), such that |E| = c|B|. Additionally, the magnetic field can be obtained using the cross product relation B = (1/c)(k? × E), where k? is the unit vector in the direction of wave propagation. This relation not only links their magnitudes but also determines the direction of the magnetic field relative to the electric field and the wave’s propagation.

Vector Cross Product in Electromagnetism

The vector cross product is a key operation in electromagnetism used to determine directions perpendicular to two given vectors. In the context of electromagnetic waves, the cross product is crucial for finding the magnetic field direction when given the electric field and the propagation direction. It ensures that the resulting field is orthogonal to both contributing vectors, consistent with the physical requirements of transverse waves.

Sinusoidal Wave Representation and Phase

Sinusoidal functions are commonly used to represent the oscillatory behavior of electromagnetic wave fields, capturing their periodic variations in both space and time. The phase of the sinusoid, which may include constant offsets, determines the initial condition or shift of the wave. This phase factor is essential for describing the temporal and spatial evolution of the wave as well as for aligning the relative timing between the electric and magnetic fields.

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