Solution

Suppose Observer $O$ sees an electron moving to the right with the speed $v=\beta c$ and the region producing the magnetic field moving to the left with the speed $v_1=\beta_1 c$. Then, Observer $O'$ moving with the region sees the electron entering the region with the speed

$$v'=\frac{v+v_1}{1+vv_1/c^2}=\frac{(\beta+\beta_1)c}{1+\beta\beta_1}.$$

Let

$$\gamma'=\frac{1}{\sqrt{1-(v'/c)^2}}=\frac{1+\beta\beta_1}{\sqrt{(1+\beta\beta_1)^2-(\beta+\beta_1)^2}}=\frac{1+\beta\beta_1}{\sqrt{(1-\beta^2)(1-\beta_1^2)}}.$$

From Problem 14.4.23, Observer $O'$ witnesses the electron moving along a semicircular arc in the region for the duration of

$$T'=\frac{\pi m_e\gamma'}{eB}=\frac{\pi m_e(1+\beta\beta_1)}{eB\cdot\sqrt{(1-\beta^2)(1-\beta_1^2)}},$$

where $B$ is the magnetic field produced in the region. As a result of time dilation, Observer $O$ records the duration as

$$T=\frac{T'}{\sqrt{1-\beta_1^2}}=\frac{\pi m_e(1+\beta\beta_1)}{eB(1-\beta_1^2)\sqrt{1-\beta^2}}.$$