Special theory of relativity > The transformation of electric and magnetic fields > Problem 14.3.15

Statement

$14.3.15$ How many times will the amplitude of a plane electromagnetic wave change
when it passes into a coordinate system moving at a speed $\beta c$ in the direction
of wave propagation?

Solution

To solve this problem we need to use the Lorentz transformation for fields, taking into account that the fields of the electromagnetic wave are
perpendicular to the velocity of the moving frame. Also, in each frame, the relation $\vec{E} = [\vec{v} \times \vec{B}]$ holds, where $\vec{v}$ is the
velocity vector of the electromagnetic wave; in this case $\vec{v} = \vec{c}$.

\begin{equation}
E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \sqrt{\frac{1-\beta}{1+\beta}}
\end{equation}

And using the relation between fields:

\begin{equation}
B' = B \sqrt{\frac{1-\beta}{1+\beta}}
\end{equation}

Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller.

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Answer

Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller.