In the experiment the following relationships are visible:
The bigger the coil current I resp. the magntic field B, the smaller the cross on the screen
The bigger the acceleration voltage Va, the bigger the cross on the screen
In some special combinations of coil current I resp. magnetic field B and acceleration voltage Va all electrons impact in a single point on the screen.
General Considerations
From a point source (the electron gun) diverging electron beams get into an uniform magnetic field, which is directed axial to the direction of the electrons. So, depending on the pitch angle $\delta$, the electrons have two velocity components - $v_{\parallel}$ parallel and $v_{\perp}$ perpendicular to the magnetic field B. They can be calculated with the initial speed $v_0$:
$$\begin{equation}v_{\parallel}=v_0\cdot \cos(\delta)\qquad \text{bzw.} \qquad v_{\perp}=v_0\cdot \sin(\delta).\end{equation}$$Motion parallel to the magnetic field
The radius of the helix is:
$$\begin{equation}r=\frac{v_{\perp}\cdot m_e}{e\cdot B}\end{equation}$$
Using $v_{\perp}=\omega\cdot r$ 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
$$\begin{equation}T=\frac{2\pi\cdot m_e}{e\cdot B}.\end{equation}$$
The orbital period T is independend from the initial speed $v_0$ and the pitch angle $\delta$. So T is equal for all electrons the they cut after T the same magentic field line as they cut when entering the magnetic field. Motion parallel to the magnetic field:
Here no forces act on the electrons and the pitch h of the trajectory can be calculated with:
$$\begin{equation}h=v_{\parallel}\cdot T\end{equation}$$
$v_{\parallel}$ depends on the pitch angle $\delta$ and so the pitch h depends also on $\delta$. As consequence for electrons with different pitch angel $\delta$ the distance between starting point P0 and intersection point Px with the starting magnetic field line is different.
This influence is minor for angles $\delta<10°$. With the small-angle approximation $v_{\parallel}=v_0$ can be assumed for all electrons. So the intersection points Px are in the same distance h from P0. The electrons are focused in one point Pfocus. The distance between P0 and Pfocus is the pitch h and can be calculated with
$$\begin{equation}h=v_{0}\cdot T =\frac{2\pi\cdot m_e\cdot v_0}{e\cdot B}\end{equation}$$
Apply on the experiment
According to (6) electrons can be focussed on a random point Pfocus with an adequate choice of Va and I resp. B. To focus the electrons on the screen in our experiment the distance between electron gun and screen must match the pitch h of the helix trajectory. In our tube the distance is hex=0,17m. So the condition for focussing the electron on the screen is:
$$\frac{v_0}{B}=\frac{h_{ex} \cdot e}{2\pi\cdot m_e}$$
Using $v_0=\sqrt{2\cdot\frac{e}{m_e}\cdot B_{\text a}}$ and using the magnetic field $B=7.48\cdot10^{-4}\frac{\text T}{\text A}\cdot I$ of our Helmholtz coils the condition can be transformed to:
$$\frac{\sqrt{V_{\text a}}}{I}=\frac{7.48\cdot 10^{-4}\cdot h_{ex}\cdot \sqrt{e}}{2\cdot\pi\cdot\sqrt{2\cdot m_e}}\approx 35.3\cdot h_{ex}\approx 6.0$$ You can test this result with the experiment on the next page.
Propagating
Motion parallel to the magnetic field