New solution

JAMF edited
View solution History
revision #18741 newer →
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]