Новое решение

JAMF правка от 16 июн. 2026 г., 18:47

правка #18940 позже →

en/12.1.30.md +31 −0

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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.
		+
		+	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]

	@@ -0,0 +1,31 @@
			### 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.

			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]