Problem — Which EM wave

Problem

Among the representations in $(x,\,t=0)$ below, identify the correct one for a plane electromagnetic wave in vacuum travelling towards $+x$, with $\vv E$ along $y$ and $\vv B$ along $z$. (a) Identify the answer remembering that $\vv E\times\vv B$ must point along $+\uv k$ at every point. (b) What does panel B describe? (c) What does panel C describe? (Inspired by Epstein 2002.)

In red $\vv E$, in blue $\vv B$.

Solution

Criterion. For a wave travelling towards $+x$ the cross product $\vv E\times\vv B$ must point towards $+\uv k$ ($+x$) at every point and instant. With $\vv E$ along $+y$ and $\vv B$ along $+z$ we have $\uv y\times\uv z=\uv x$: the two fields must therefore be in phase (positive together, negative together).

(a) Correct answer. In panel A the fields $\vv E$ and $\vv B$ are in phase: where one is positive so is the other, so $\vv E\times\vv B$ always points towards $+x$.

$\ev{\text{A}}$

(b) Panel B. Here $\vv B$ is $90^\circ$ out of phase with $\vv E$ (a cosine versus a sine). It does not represent a travelling plane wave: Maxwell’s equations require $\vv E$ and $\vv B$ to be in phase, so B is not physically valid.

(c) Panel C. Here $\vv B$ is in antiphase with $\vv E$ (opposite sign everywhere). Then $\vv E\times\vv B$ points towards $-x$: it describes a wave travelling backwards (towards $-x$), not the one required.

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Topics: Onde elettromagnetiche Concepts: Onda elettromagnetica · Vettore di Poynting