# Evaluation and Explanation of the Experiment

## Observations

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 $v_0$: $$$$v_{\parallel}=v_0\cdot \cos(\delta)\qquad \text{bzw.} \qquad v_{\perp}=v_0\cdot \sin(\delta).$$$$ Motion parallel to the magnetic field
The radius of the helix is: $$$$r=\frac{v_{\perp}\cdot m_e}{e\cdot B}$$$$ Using $v_{\perp}=\omega\cdot r$ the angular velocity $\omega$ is $$$$\omega=\frac{e\cdot B}{m_e}$$$$ and with $\omega=\frac{2\pi}{T}$ the orbital period T of the electrons is $$$$T=\frac{2\pi\cdot m_e}{e\cdot B}.$$$$ The orbital period T is independend from the initial speed $v_0$ and the pitch angle $\delta$. So T is equal for all electrons the they cut after T the same magentic field line as they cut when entering the magnetic field.
Motion parallel to the magnetic field:
Here no forces act on the electrons and the pitch h of the trajectory can be calculated with: $$$$h=v_{\parallel}\cdot T$$$$ $v_{\parallel}$ depends on the pitch angle $\delta$ and so the pitch h depends also on $\delta$. As consequence for electrons with different pitch angel $\delta$ the distance between starting point P0 and intersection point Px with the starting magnetic field line is different.
This influence is minor for angles $\delta<10°$. With the small-angle approximation $v_{\parallel}=v_0$ can be assumed for all electrons. So the intersection points Px are in the same distance h from P0. The electrons are focused in one point Pfocus. The distance between P0 and Pfocus is the pitch h and can be calculated with $$$$h=v_{0}\cdot T =\frac{2\pi\cdot m_e\cdot v_0}{e\cdot B}$$$$

## 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 $v_0=\sqrt{2\cdot\frac{e}{m_e}\cdot U_{\text b}}$ 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: $$\frac{\sqrt{U_{\text b}}}{I}=\frac{7{,}48\cdot 10^{-4}\cdot h_{ex}\cdot \sqrt{e}}{2\cdot\pi\cdot\sqrt{2\cdot m_e}}\approx 35{,}3\cdot h_{ex}\approx 6{,}0$$ You can test this result with the experiment on the next page.