Правка разделов «Statement», «Solution», «Answer»
en/14.3.15.md
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| ### Statement | |||
| − | $14.3.15.$ [Insert the problem statement] | ||
| − | |||
| − | ### Solution | ||
| − | |||
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| − | \begin{document} | ||
| $14.3.15$ 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? | ||
| + | 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. | |||
| \end{document} | |||
| @@ -50,4 +26,4 @@Solution | |||
| #### Answer | |||
| − | |||
| + | Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | ||
| @@ -1,35 +1,11 @@ | |||
| ### Statement | ### Statement | ||
| $14.3.15.$ [Insert the problem statement] | |||
| ### Solution | |||
| \documentclass[12pt,a4paper]{article} | |||
| \usepackage[english]{babel} | |||
| \usepackage{float} | |||
| \usepackage{wrapfig} | |||
| \usepackage{lmodern} | |||
| \usepackage[T1]{fontenc} | |||
| \usepackage[utf8]{inputenc} | |||
| \usepackage{microtype} | |||
| \usepackage{graphicx} | |||
| \usepackage{booktabs} | |||
| \usepackage{amsmath,amssymb} | |||
| \usepackage{hyperref} | |||
| \usepackage{csquotes} | |||
| \usepackage{geometry} | |||
| \usepackage{subcaption} | |||
| \usepackage{tikz} | |||
| \usepackage{array} | |||
| \usepackage{pgfplots} | |||
| \usepackage{wrapfig} | |||
| \usepackage{subcaption} | |||
| \begin{document} | |||
| $14.3.15$ 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 | |||
| 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 | ||
| 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-\beta}{1+\beta}} | E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \sqrt{\frac{1-\beta}{1+\beta}} | ||
| \end{equation} | \end{equation} | ||
| And using the relation between fields: | And using the relation between fields: | ||
| \begin{equation} | \begin{equation} | ||
| B' = B \sqrt{\frac{1-\beta}{1+\beta}} | 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} | ||
| @@ -50,4 +26,4 @@Solution | |||
| #### Answer | #### Answer | ||
| Thus, the amplitude becomes $\sqrt{\frac{1-\beta}{1+\beta}}$ times smaller. | |||