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GATE 2021 PH – Question 12

Electromagnetic theory · Electromagnetic waves in free space, non-conducting and conducting media · 1 mark · Multiple choice

A matter wave is represented by $\Psi(x,y,z,t)=A\exp[i(4x+3y+5z-10\pi t)]$, where $A$ is constant. The unit vector along its propagation direction is

  1. $\frac4{5\sqrt2}\hat{\mathbf x}+\frac3{5\sqrt2}\hat{\mathbf y}+\frac1{\sqrt2}\hat{\mathbf z}$
  2. $\frac3{5\sqrt2}\hat{\mathbf x}+\frac4{5\sqrt2}\hat{\mathbf y}+\frac1{5\sqrt2}\hat{\mathbf z}$
  3. $\frac1{5\sqrt2}\hat{\mathbf x}+\frac3{5\sqrt2}\hat{\mathbf y}+\frac1{\sqrt2}\hat{\mathbf z}$
  4. $\frac1{5\sqrt2}\hat{\mathbf x}+\frac4{5\sqrt2}\hat{\mathbf y}+\frac3{5\sqrt2}\hat{\mathbf z}$

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Correct answer: (A) $\frac4{5\sqrt2}\hat{\mathbf x}+\frac3{5\sqrt2}\hat{\mathbf y}+\frac1{\sqrt2}\hat{\mathbf z}$

Explanation

The spatial phase identifies $\mathbf k=4\hat{\mathbf x}+3\hat{\mathbf y}+5\hat{\mathbf z}$. Its magnitude is $\sqrt{4^2+3^2+5^2}=5\sqrt2$. Divide each component by this magnitude to obtain the unit vector in A. The sign of $\mathbf k\cdot\mathbf r-\omega t$ gives propagation along $+\mathbf k$.

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