Write down the (real) electric and magnetic fields for a...

Write down the (real) electric and magnetic fields for a monochromatic plane wave of amplitude $E_{0},$ frequency $\omega,$ and phase angle zero that is (a) traveling in the negative $x$ direction and polarized in the $z$ direction; (b) traveling in the direction from the origin to the point $(1,1,1),$ with polarization parallel to the $x z$ plane. In each case, sketch the wave, and give the explicit Cartesian components of $\mathbf{k}$ and $\hat{\mathbf{n}}$.

Key Concepts

Polarization and Cartesian Component Analysis

Polarization refers to the orientation of the oscillations of the electric field in a plane wave. It is described by a unit vector that specifies the direction of the field's oscillation, and it must be perpendicular to the wave vector. Expressing the polarization and the wave vector in Cartesian components facilitates precise descriptions of the fields in specific directions, such as when aligning them with given coordinate axes or planes.

Magnetic Field Representation

Associated with the electric field in a plane electromagnetic wave is the magnetic field, which is also sinusoidally varying and perpendicular to both the electric field and the direction of propagation. Its magnitude and phase are related to those of the electric field, often derived using the relation B = (1/c) k? × E, ensuring that both fields satisfy Maxwell's equations and maintain a consistent phase relationship.

Wave Vector (k) and Propagation Direction

The wave vector, denoted as k, describes the direction in which the plane wave travels and is defined by its Cartesian components. Its magnitude is related to the wavelength (or frequency) of the wave, and its direction determines the orientation of the wavefronts. Analyzing k in Cartesian coordinates allows for explicit description of the propagation direction in problems involving geometrical orientations.

Electric Field Representation

The electric field of a plane wave is typically represented as a sinusoidally varying vector function that encapsulates the amplitude, frequency, and phase of the wave. It is written in a form such as E(r,t) = E0 cos(k·r - ?t + ?) multiplied by a unit vector defining its polarization, indicating that its magnitude and direction may be modulated according to the propagation direction and polarization requirements.

Electromagnetic Plane Waves

Electromagnetic plane waves are solutions to Maxwell's equations in free space, characterized by electric and magnetic fields that oscillate sinusoidally in space and time. They propagate in a fixed direction with their wavefronts being infinite planes, and the fields are mutually perpendicular as well as perpendicular to the direction of propagation.

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