Правка разделов «Statement», «Solution»
en/12.1.30.md
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| ### Statement | |||
| − | $12.1.30.$ [Insert the problem statement] | ||
| − | |||
| − | ### Solution | ||
| − | |||
| $12.1.30$ The amplitude of the wave when reflected from a metal wall moving towards | |||
| it increases by a factor of $k$. Determine the wall speed. | |||
| + | |||
| + | ### Solution | ||
| Using the relation between the magnetic and electric fields of an electromagnetic wave $E = B c$, and the | |||
| Lorentz transformation for electric and magnetic fields: | |||
| \begin{equation} | |||
| E = \frac{E_0 - (-v) E_0/c}{\sqrt{1-v^2/c^2}} = E_0 \sqrt{\frac{1+v/c}{1-v/c}} | |||
| \end{equation} | |||
| And finally the amplitude of the reflected wave in the Earth frame is: | |||
| \begin{equation} | |||
| E_1 = E_0 \frac{1+v/c}{1-v/c} = E_0 k | |||
| \end{equation} | |||
| Then, the velocity of the wall is: | |||
| \begin{equation} | |||
| v = c \frac{k - 1}{k + 1} | |||
| \end{equation} | |||
| #### Answer | |||
| [Insert a concise answer or boxed result] | |||
| ещё строк без изменений 20 | |||
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| ### Statement | ### Statement | ||
| $12.1.30.$ [Insert the problem statement] | |||
| ### Solution | |||
| $12.1.30$ The amplitude of the wave when reflected from a metal wall moving towards | $12.1.30$ The amplitude of the wave when reflected from a metal wall moving towards | ||
| it increases by a factor of $k$. Determine the wall speed. | it increases by a factor of $k$. Determine the wall speed. | ||
| ### Solution | |||
| Using the relation between the magnetic and electric fields of an electromagnetic wave $E = B c$, and the | Using the relation between the magnetic and electric fields of an electromagnetic wave $E = B c$, and the | ||
| Lorentz transformation for electric and magnetic fields: | Lorentz transformation for electric and magnetic fields: | ||
| \begin{equation} | \begin{equation} | ||
| E = \frac{E_0 - (-v) E_0/c}{\sqrt{1-v^2/c^2}} = E_0 \sqrt{\frac{1+v/c}{1-v/c}} | E = \frac{E_0 - (-v) E_0/c}{\sqrt{1-v^2/c^2}} = E_0 \sqrt{\frac{1+v/c}{1-v/c}} | ||
| \end{equation} | \end{equation} | ||
| And finally the amplitude of the reflected wave in the Earth frame is: | And finally the amplitude of the reflected wave in the Earth frame is: | ||
| \begin{equation} | \begin{equation} | ||
| E_1 = E_0 \frac{1+v/c}{1-v/c} = E_0 k | E_1 = E_0 \frac{1+v/c}{1-v/c} = E_0 k | ||
| \end{equation} | \end{equation} | ||
| Then, the velocity of the wall is: | Then, the velocity of the wall is: | ||
| \begin{equation} | \begin{equation} | ||
| v = c \frac{k - 1}{k + 1} | v = c \frac{k - 1}{k + 1} | ||
| \end{equation} | \end{equation} | ||
| #### Answer | #### Answer | ||
| [Insert a concise answer or boxed result] | [Insert a concise answer or boxed result] | ||
| ещё строк без изменений 20 | |||