Solution

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$14.4.22$ 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}

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