Mathematical view of the helix trajectory

For a mathematical view of the helix trajectory perpendicular and parallel motion are observed separately. Therefor the initial speed v0 is seperated with trigonometric functions in $v_{\parallel}$ and $v_{\perp}$.Schemtic view of helix trajectory of electrons in magnetic fieldMotion perpendicular to the magnetic field:
Here the same equations as in the experiment with the cathode ray tubes can be used: $$\begin{equation}F_{\rm{Lorentz}}=F_{\rm{Zentripetal}}\qquad \Rightarrow\qquad e\cdot v_{\perp}\cdot B=m_e\frac{v_{\perp}^2}{r}\end{equation}$$ So the radius of the helix is: $$\begin{equation}r=\frac{v_{\perp}\cdot m_e}{e\cdot B}\end{equation}$$ With $v_{\perp}=\omega\cdot r$ the angular velocity $\omega$ is: $$\begin{equation}\omega=\frac{e\cdot B}{m_e}\end{equation}$$ and with $\omega=\frac{2\pi}{T}$ the orbital period T of the electrons is $$\begin{equation}T=\frac{2\pi\cdot m_e}{e\cdot B}.\end{equation}$$ Motion parallel to the magnetic field:
Here no forces act on the electrons. The distance parallel to the magnetic field which the electrons cover while fulfilling a complete rotation is called the pitch p of the helix. It can be calculated with: $$p=v_{\parallel}\cdot T=\frac{2\pi\cdot m_e \cdot v_{\parallel}}{e\cdot B}$$