The magnetic field of an EM wave is given by By= Bm sin(k z+ωt), Bx=0, and Bz=0 . (a) In what di

step 1 Now bm sine k z plus omega t points towards negative z direction and e cross b gives the directi

The magnetic field of an EM wave is given by By= Bm sin(k...

The magnetic field of an EM wave is given by $B_{y}=$ $B_{\mathrm{m}} \sin (k z+\omega t), B_{x}=0,$ and $B_{z}=0 .$ (a) In what direction is this wave traveling? (b) Write expressions for the components of the electric field of this wave.

Key Concepts

Orthogonality of Field Components

In an electromagnetic plane wave, the electric field (E), the magnetic field (B), and the direction of propagation (given by the wave vector k) are mutually perpendicular. This transverse nature stems from Maxwell’s equations in a source?free region and ensures that energy propagates without any longitudinal field components. This orthogonality also forms the basis of the right-hand rule used to determine the direction of one field given the directions of the others.

Electromagnetic Plane Waves

Electromagnetic plane waves are solutions to Maxwell’s equations in free space, where both the electric field and the magnetic field oscillate sinusoidally, have consistent phase relationships, and propagate in a specific direction. These waves are characterized by the fact that the field amplitudes remain constant on planes perpendicular to the direction of propagation, making analyses and derivations of field components straightforward.

Phase and Propagation Direction

The argument of the sinusoidal function in the wave equation, often written as (k·r ± ?t), determines the wave’s phase. The sign between the spatial part (k·r) and the temporal part (?t) indicates the direction of propagation. A positive sign implies that the phase constant surfaces move in the direction opposite to the wave vector (i.e., along -k?), while a negative sign indicates propagation in the direction of k?. Understanding this relationship is crucial for determining how the wave travels through space.

Electric and Magnetic Field Relationships

Maxwell’s curl equations govern the relationship between the electric and magnetic fields in an electromagnetic wave. Specifically, the cross product of the wave vector and the electric field is related to the magnetic field (k × E = ?B), which implies that the magnitudes of these fields are connected by the phase velocity of the wave (in free space, c = ?/k). This relation allows one to derive expressions for one field component in terms of the other, ensuring that both fields are in phase and correctly oriented with respect to the direction of propagation.

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