Which forces acts on the electrons in the b-field of two Helmholtz coils?

The Lorentz force is acting on moving charges in a magnetic field:$$\vec F_{\rm{Lorentz}} = q\cdot \vec v_0 \times \vec{B}$$

The Lorentz force acting on electrons in magnetic fields directs in the center of an circular path

Because of the Electrons move always perpendicular to the magnetic field ($\vec v_0$ perpendicular to $\vec B$) and charge $q$ is the charge of an electron the equation can be simplified to$$ F_{\rm{Lorentz}} = e\cdot v_0 \cdot B$$ You can get the radius because the Lorentz force is here the centripetal force $F_{\rm{Zentripetal}} = m_e\frac{{v_0}^2}{r}$, with the electron mass $m_e$ and the radius $r$. $$F_{\rm{Lorentz}}= F_{\rm{Zentripetal}}\Leftrightarrow e\cdot v_0 \cdot B = m_e\frac{{v_0}^2}{r}$$ Solved for $r$ the radius of the electron-trajectory, also called Larmor radius or cyclotron radius, is:$$\bbox[7px,border:2px solid red] {r = \frac {m_e\cdot v_0}{e\cdot B}}$$