Special theory of relativity > Motion of relativistic particles in electric and magnetic fields > Problem 14.4.22

Statement

$14.4.22$ 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?

Solution

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 \cdot \sqrt{1000^2 - 1}} \approx 0.28 \mu\text{T}
\end{equation}

Answer

\begin{equation}
B \approx 0.28 \mu\text{T}
\end{equation}