- #1
christianpoved
- 15
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Homework Statement
To solve the wave equations in vacuum for ##\vec{E}## and ##\vec{B}## we made the ansatz:
\begin{array}{cc}
\vec{E}\left(\vec{r},t\right)=\vec{E}_{0}\cos\left(\vec{k}\cdot\vec{r}-\omega t+\delta\right)
\vec{B}\left(\vec{r},t\right)=\vec{B}_{0}\cos\left(\vec{k}\cdot\vec{r}-\omega t+\delta\right)
\end{array}
Make a new ansatz of the form:
\begin{array}{cc}
\vec{E}\left(\vec{r},t\right)=\vec{E}_{0}\cos\left(\vec{k}\cdot \vec{r}-\omega t+\delta\right)
\vec{B}\left(\vec{r},t\right)=\vec{B}_{0}\cos\left(\vec{k}^{\prime} \cdot \vec{r}-omega^{\prime}t+\delta^{\prime}\right)
\end{array}
And show either that
\begin{array}{cc}
\vec{k}=\vec{k}^{\prime}
\omega=\omega^{\prime}
\end{array}
Or that, in general:
\begin{array}{cc}
\vec{k}^{\prime}\left(\omega^{\prime}\right)=\vec{k}\left(\omega\right)
\end{array}
Homework Equations
I used Maxwell equations and some cross/dot product identities.
The Attempt at a Solution
I put these new fields into Maxwell equations and i get this
\begin{array}{cc}
\vec{E}\cdot\vec{k}=0 & \mbox{(1)}\\
\vec{B}\cdot\vec{k}^{\prime}=0 & \mbox{(2)}\\
\vec{k}\times\vec{E}=\omega^{\prime}\vec{B} & \mbox{(3)}\\
\vec{k}^{\prime}\times\vec{B}=-\frac{\vec{k}\cdot\vec{k}}{\omega}\vec{E} & \mbox{(4)}
\end{array}
With these equations I can show that both wavenumbers are parallel, and that the factor between them is the ratio of the frequencies, shouldn't I be able to show that the frequencies are the same and then conclude that the wavenumbers are equal too?
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