The electric field of a monochromatic plane wave travelling in a lossless isotropic and homogenous medium is given by E(z, t) = ...
Q. The electric field of a monochromatic plane wave travelling in a lossless isotropic and homogenous medium is given by E(z, t) = E₀ [x̂ cos(ωt − kz) + Å· sin(ωt − kz)] in a right-handed orthogonal co-ordinate system. Which of the following is the correct polarization of the electromagnetic wave?
(A) Right-handed circularly polarized
Correct answer under the IEEE convention.
(B) Left-handed circularly polarized
(C) Linearly polarized
(D) Linearly polarized at −45° to x̂
Solution
1. Identify the electric-field components
Let ψ = ωt − kz. The components along the x and y directions are:
Eâ‚“ = E₀ cos ψ
Eᵧ = E₀ sin ψ
Both components have equal amplitudes, the same frequency, and a phase difference of 90°. These are the conditions for circular polarization.
2. Verify the magnitude of the electric field
|E⃗| = √(Eâ‚“² + Eᵧ²)
Substituting the components:
|E⃗| = E₀ √(cos² ψ + sin² ψ)
|E⃗| = E₀
Since cos² ψ + sin² ψ = 1, the magnitude remains constant. The electric-field vector rotates in the xy-plane, confirming circular polarization.
3. Determine the handedness
The phase is ψ = ωt − kz, so the wave propagates in the +z direction. At a fixed position, z = 0:
At t = 0: E⃗ = E₀ x̂
For a small positive time: Eₓ decreases while Eᵧ becomes positive.
Therefore, the field rotates from +x̂ toward +Å· as time increases.
Under the IEEE convention, this rotation for propagation in the +z direction corresponds to right-handed circular polarization.
Final answer
Option (A) — Right-handed circularly polarized
Summary
- Equal amplitudes and a 90° phase difference indicate circular polarization.
- Unequal amplitudes with a 90° phase difference generally indicate elliptical polarization.
- Components in phase or 180° out of phase indicate linear polarization.
- The handedness depends on the propagation direction, phase convention, and the polarization convention being used.