Новое решение

JAMF правка от 13 июн. 2026 г., 20:00

правка #18731 позже →

en/14.3.16.md +32 −0

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		+	### Statement
		+
		+	$14.3.16.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	$14.3.16$ Solve problem $14.3.15$ for plane wave propagation in a dielectric medium with
		+	refractive index $n$.
		+
		+	$14.3.15$ 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?
		+
		+	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 $\vec{v} = \frac{c}{n}$.
		+
		+	\begin{equation}
		+	E' = \frac{E - \beta c B}{\sqrt{1-\beta^2}} = E \frac{1-\beta n}{\sqrt{1-\beta^2}}
		+	\end{equation}
		+
		+	And using the relation between fields:
		+
		+	\begin{equation}
		+	B' = B \frac{1-\beta n}{\sqrt{1-\beta^2}}
		+	\end{equation}
		+
		+	So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller.
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

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			### Statement

			$14.3.16.$ [Insert the problem statement]

			### Solution

			$14.3.16$ Solve problem $14.3.15$ for plane wave propagation in a dielectric medium with
			refractive index $n$.

			$14.3.15$ 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?

			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 $\vec{v} = \frac{c}{n}$.

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

			And using the relation between fields:

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

			So, the amplitude becomes $\frac{1-\beta n}{\sqrt{1-\beta^2}}$ times smaller.

			#### Answer

			[Insert a concise answer or boxed result]