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) + ŷ 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:

Eₓ = 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| = √(Eₓ² + 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 +ŷ 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.