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GATE 2024 EC – Question 39

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

A uniform plane wave with electric field $E(x)=A_y\hat a_ye^{-j\frac{2\pi x}{3}}$ V/m is travelling in the air (relative permittivity, $\epsilon_r=1$ and relative permeability, $\mu_r=1$) in the $+x$ direction ($A_y$ is a positive constant, $\hat a_y$ is the unit vector along $y$ axis). It is incident normally on an ideal electric conductor (conductivity, $\sigma=\infty$) at $x=0$. The position of the first null of the total magnetic field in the air (measured from $x=0$, in metres) is ______.

  1. $-\dfrac34$
  2. $-\dfrac32$
  3. $-6$
  4. $-3$

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Correct answer: (A) $-\dfrac34$

Explanation

The phase constant is $k=2\pi/3$ rad/m, so $\lambda=3$ m. For a perfect conductor the reflected $E$ has coefficient $-1$ and the reflected $H$ has $+1$. The total magnetic field is $H\propto e^{-jkx}+e^{jkx}=2\cos kx$, which is zero when $kx=-\pi/2$, i.e. $x=-\lambda/4=-\tfrac34$ m.