(II) The electric field of a plane EM wave is given by Ex=...

(II) The electric field of a plane EM wave is given by $E_{x}=$ $E_{0} \cos (k z+\omega t), \quad E_{y}=E_{z}=0,$ Determine
(a) the direction of propagation and $(b)$ the magnitude and direction of $\overrightarrow{\mathbf{B}}$.

Key Concepts

Plane Electromagnetic Waves

Plane electromagnetic waves are solutions to Maxwell's equations in free space where the electric and magnetic fields oscillate sinusoidally and propagate in a specific direction. In such waves, the fields are uniform (or planar) along planes perpendicular to the direction of propagation, meaning that any variation occurs only along one spatial dimension as a function of time.

Wave Vector and Propagation Direction

The wave vector describes the direction in which the phase of the wave advances in space. In the plane wave equation, the argument of the cosine function combines the spatial and temporal components (e.g., kz + ?t), thereby indicating that the wave propagates in the direction associated with the spatial coordinate present in the phase. This concept allows us to determine the propagation direction by scrutinizing the phase expression.

Relationship between Electric and Magnetic Fields

In electromagnetic waves, the electric and magnetic fields are intrinsically linked by Maxwell's equations. This relationship ensures that the magnetic field is not only perpendicular to the electric field but also has a magnitude proportional to it. Typically, the magnetic field magnitude is given by B = E/c, where c is the speed of light in vacuum, signifying that the two fields are in phase and their oscillations are coupled as the wave propagates.

Right-Hand Rule

The right-hand rule is a mnemonic that helps in identifying the orientation of the magnetic field given the direction of the electric field and the propagation direction of the wave. According to this rule, if you point the fingers of your right hand in the direction of the electric field and curl them toward the propagation direction, your thumb will point in the direction of the magnetic field. This concept is essential for determining the spatial orientation of the fields in a propagating electromagnetic wave.

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