Новое решение

JAMF правка от 14 июн. 2026 г., 03:34

правка #18743 позже →

en/14.4.22.md +57 −0

		@@ -0,0 +1,57 @@
		+	### Statement
		+
		+	$14.4.22.$ [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}
		+
		+	$14.4.22$ What is the magnetic field induction on storage tracks of radius R = 6 m, if
		+	the mass of electrons moving along these tracks is N = 1000 times greater
		+	than me?
		+
		+	In this problem it is necessary to use the concept of relativistic mass. The relativistic mass of each electron in this case is:
		+
		+	\begin{equation}
		+	M = m_e N = \frac{m_e}{\sqrt{1-\beta^2}} \rightarrow \beta = \frac{\sqrt{N^2-1}}{N}
		+	\end{equation}
		+
		+	Thus, the velocity of the electrons inside the storage is:
		+
		+	\begin{equation}
		+	v = \beta c = c \frac{\sqrt{N^2 - 1}}{N}
		+	\end{equation}
		+
		+	On the other hand, the radial force is equal to the magnetic force acting on the electron $F = e v B = e c B \frac{\sqrt{N^2 - 1}}{N}$. Using Newton's second law
		+	we can calculate the magnetic field induction inside the storage:
		+
		+	\begin{equation}
		+	\frac{N m_e v^2}{R} = e v B \rightarrow B = \frac{N m_e v}{e R} = \frac{m_e c}{e R} \sqrt{N^2 - 1} = \frac{m_e c}{e \cdot 6} \sqrt{1000^2 - 1} \approx 1.7 \, \text{T}
		+	\end{equation}
		+
		+	\end{document}
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

	@@ -0,0 +1,57 @@
			### Statement

			$14.4.22.$ [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}

			$14.4.22$ What is the magnetic field induction on storage tracks of radius R = 6 m, if
			the mass of electrons moving along these tracks is N = 1000 times greater
			than me?

			In this problem it is necessary to use the concept of relativistic mass. The relativistic mass of each electron in this case is:

			\begin{equation}
			M = m_e N = \frac{m_e}{\sqrt{1-\beta^2}} \rightarrow \beta = \frac{\sqrt{N^2-1}}{N}
			\end{equation}

			Thus, the velocity of the electrons inside the storage is:

			\begin{equation}
			v = \beta c = c \frac{\sqrt{N^2 - 1}}{N}
			\end{equation}

			On the other hand, the radial force is equal to the magnetic force acting on the electron $F = e v B = e c B \frac{\sqrt{N^2 - 1}}{N}$. Using Newton's second law
			we can calculate the magnetic field induction inside the storage:

			\begin{equation}
			\frac{N m_e v^2}{R} = e v B \rightarrow B = \frac{N m_e v}{e R} = \frac{m_e c}{e R} \sqrt{N^2 - 1} = \frac{m_e c}{e \cdot 6} \sqrt{1000^2 - 1} \approx 1.7 \, \text{T}
			\end{equation}

			\end{document}

			#### Answer

			[Insert a concise answer or boxed result]