Новое решение

JAMF правка от 14 июн. 2026 г., 01:26

правка #18741 позже →

en/14.4.23.md +37 −0

		@@ -0,0 +1,37 @@
		+	### Statement
		+
		+	$14.4.23.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	$14.4.23$ An electron enters a magnetic field at a velocity $\beta c$ perpendicular to the
		+	field boundary and to the induction vector $B$. Determine the residence time of
		+	the electron in the magnetic field.
		+
		+	When the electron enters the magnetic field, it will describe a circular path of radius $R$.
		+	To calculate the radius we need to use Newton's second law for the radial direction.
		+
		+	\begin{equation}
		+	\frac{m_e v^2}{R} = \frac{m_e (\beta c)^2}{R} = F
		+	\end{equation}
		+
		+	The force acting on the electron is the magnetic force $F = e [\vec{v} \times \vec{B}]$; because the magnetic field is perpendicular to the velocity of the
		+	electron we have $F = e \beta c B$. Finally:
		+
		+	\begin{equation}
		+	\frac{m_e (\beta c)^2}{R} = e \beta c B \rightarrow R = \frac{m_e \beta c}{e B}
		+	\end{equation}
		+
		+	Thus, the time the electron spends in the magnetic field is:
		+
		+	\begin{equation}
		+	t = \frac{2 \pi R}{\beta c} = \frac{2 \pi m_e}{e B}
		+	\end{equation}
		+
		+	\begin{equation}
		+	t = \frac{2 \pi m_e}{e B}
		+	\end{equation}
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

	@@ -0,0 +1,37 @@
			### Statement

			$14.4.23.$ [Insert the problem statement]

			### Solution

			$14.4.23$ An electron enters a magnetic field at a velocity $\beta c$ perpendicular to the
			field boundary and to the induction vector $B$. Determine the residence time of
			the electron in the magnetic field.

			When the electron enters the magnetic field, it will describe a circular path of radius $R$.
			To calculate the radius we need to use Newton's second law for the radial direction.

			\begin{equation}
			\frac{m_e v^2}{R} = \frac{m_e (\beta c)^2}{R} = F
			\end{equation}

			The force acting on the electron is the magnetic force $F = e [\vec{v} \times \vec{B}]$; because the magnetic field is perpendicular to the velocity of the
			electron we have $F = e \beta c B$. Finally:

			\begin{equation}
			\frac{m_e (\beta c)^2}{R} = e \beta c B \rightarrow R = \frac{m_e \beta c}{e B}
			\end{equation}

			Thus, the time the electron spends in the magnetic field is:

			\begin{equation}
			t = \frac{2 \pi R}{\beta c} = \frac{2 \pi m_e}{e B}
			\end{equation}

			\begin{equation}
			t = \frac{2 \pi m_e}{e B}
			\end{equation}

			#### Answer

			[Insert a concise answer or boxed result]