Edit to “Solution”
en/14.3.15.md
+4 −4
| ### 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 | |||
| @@ -11,13 +11,13 @@Solution | |||
| 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 | ||
| + | E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \sqrt{\frac{1+\beta}{1-\beta}} | ||
| \end{equation} | |||
| + | We have take into account $[\vec{c} \times \vec{B}]$ | ||
| + | is antiparallel with $\vec{E}$. And using the relation between fields: | ||
| − | And using the relation between fields: | ||
| − | |||
| \begin{equation} | |||
| − | B' = B \sqrt{\frac{1 | ||
| + | B' = B \sqrt{\frac{1+\beta}{1-\beta}} | ||
| \end{equation} | |||
| Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | |||
| \end{document} | |||
| #### Answer | |||
| Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | |||
| unchanged lines 6 | |||
| ### Statement | ### Statement | ||
| $14.3.15$ How many times will the amplitude of a plane electromagnetic wave change | $14.3.15$ 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 | when it passes into a coordinate system moving at a speed $\beta c$ in the direction | ||
| of wave propagation? | of wave propagation? | ||
| ### Solution | ### 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 | 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 | 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 | ||
| @@ -11,13 +11,13 @@Solution | |||
| velocity vector of the electromagnetic wave; in this case $\vec{v} = \vec{c}$. | velocity vector of the electromagnetic wave; in this case $\vec{v} = \vec{c}$. | ||
| \begin{equation} | \begin{equation} | ||
| E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \sqrt{\frac{1 |
E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \sqrt{\frac{1+\beta}{1-\beta}} | ||
| \end{equation} | \end{equation} | ||
| We have take into account $[\vec{c} \times \vec{B}]$ | |||
| is antiparallel with $\vec{E}$. And using the relation between fields: | |||
| And using the relation between fields: | |||
| \begin{equation} | \begin{equation} | ||
| B' = B \sqrt{\frac{1 |
B' = B \sqrt{\frac{1+\beta}{1-\beta}} | ||
| \end{equation} | \end{equation} | ||
| Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | ||
| \end{document} | \end{document} | ||
| #### Answer | #### Answer | ||
| Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | ||
| unchanged lines 6 | |||