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GATE 2017 EC – Question 64

Electromagnetics · Plane Waves and Properties · 2 marks · Multiple choice

The expression for an electric field in free space is $\mathbf{E} = E_0(\hat{x} + \hat{y} + j2\hat{z})e^{-j(\omega t - kx + ky)}$, where $x, y, z$ represent the spatial coordinates, $t$ represents time, and $\omega, k$ are constants. This electric field

  1. does not represent a plane wave.
  2. represents a circularly polarized plane wave propagating normal to the $z$-axis.
  3. represents an elliptically polarized plane wave propagating along the $x$-$y$ plane.
  4. represents a linearly polarized plane wave.

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Correct answer: (C) represents an elliptically polarized plane wave propagating along the $x$-$y$ plane.

Explanation

The phase depends only on $x - y$, so this is a plane wave travelling along the direction $(1, -1, 0)$, which lies in the $x$-$y$ plane. The field $\hat{x} + \hat{y} + j2\hat{z}$ is perpendicular to that direction. It splits into two perpendicular parts, one along $(1, 1, 0)$ with magnitude $\sqrt{2}$ and one along $\hat{z}$ with magnitude 2. They are 90° out of phase and have unequal amplitudes, so the wave is elliptically polarized.