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Which of the following electric field and magnetic field represent a propagating electromagnetic wave ? (\(\hat{k}\) is a unit vector in the direction of the propagation).

  • A.\(\vec{E} = \vec{E_{0}}(\hat{k}\cdot\vec{r} - ct)^{2}\) \(e^{-(\hat{k}.\vec{r} - ct)^{2}}\), \(\vec{B} = \hat{k} \times \frac{\vec{E_{0}}}{c}(\hat{k}\cdot\vec{r} - ct)^{2}\) \(e^{-(\hat{k}.\vec{r} - ct)^{2}}\)
  • B.\(\vec{E} = \vec{E_{0}}\sin(\hat{k}\cdot\vec{r} - ct), \vec{B} = \vec{0}\)
  • C.\(\vec{E} = \vec{E_{0}}\sin(\hat{k}\cdot\vec{r})\cos(ct)\), \(\vec{B} = \hat{k} \times \frac{\vec{E_{0}}}{c}\sin(\hat{k}\cdot\vec{r})\cos(ct)\)
  • D.\(\vec{E} = (\hat{k}\cdot\vec{r})e^{-i(\hat{k}.\vec{r} - ct)}\), \(\vec{B} = \frac{\vec{E_{0}}}{c}(\hat{k}\cdot\vec{r})e^{-i(\hat{k}.\vec{r} - ct)}\)

Answer: A

The pair that represents a propagating wave is the Gaussian-shaped pulse in \(\hat{k}\cdot\vec{r} - ct\), with \(\vec{B} = \hat{k}\times\vec{E}/c\).

A travelling wave in vacuum is any function of the single combination \(\hat{k}\cdot\vec{r} - ct\), so it moves along \(\hat{k}\) at speed \(c\) without changing shape, and its magnetic field is \(\vec{B} = \hat{k}\times\vec{E}/c\), perpendicular to both \(\vec{E}\) and \(\hat{k}\). The pulse in the keyed option has exactly this form (it need not be sinusoidal).

The others fail. A wave with \(\vec{B} = 0\) cannot satisfy Faraday's law, since a changing \(\vec{E}\) needs a \(\vec{B}\). The product \(\sin(\hat{k}\cdot\vec{r})\cos(ct)\) is a standing wave, which does not propagate. The last option's \(\vec{E}\) is not a function of \(\hat{k}\cdot\vec{r} - ct\) alone, because of the extra factor \(\hat{k}\cdot\vec{r}\), and its \(\vec{B}\) is not \(\hat{k}\times\vec{E}/c\).

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