You could be accelerating electrons to non-relativistic velocities after a trip to Home Depot and a visit to your local AC repair store and neon sign shop! Best of all, no advanced electrical or mechanical knowledge or tools are required.
Judges, see step 12 for contest entry details.
Here's some video evidence:
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Step 1: Design/Theory
Essentially, our cathode ray tube is just two electrodes in a vacuum chamber with a high voltage applied between them. When enough of the air in the chamber has been removed, electrons will freely accelerate from the negative electrode (cathode) towards the positive electrode (anode). However, instead of impacting the anode and returning to the power supply, some electrons will fly right past it and keep going until they hit a glass wall.
Some interesting effects that can be observed at this stage are sputtering and magnetic deflection.
If the acceleration potential is high enough, then some electrons striking the anode will have enough energy to knock metal ions right off the electrode. These ions will be deposited on the walls of the chamber near the anode and will create a silvery band somewhat reminiscent of the "getter" inside of an old vacuum tube.
In physics, we all learned the Lorentz force law ( F = q[E + v x B] ), or the force on a point charge due to electromagnetic fields. In this context, it tells us that electrons will be accelerated from the cathode to the anode due E, the electrostatic field created by the high voltage power supply and that those electrons will also be accelerated by another field, B, in a manner that is dependent on the velocity, v, of those electrons. Since the velocity vectors of the electrons will be pointing roughly from the cathode to the the anode without an external magnetic field, we can use this to find out what effect a magnetic field will have if we introduce one.
Let's say we bring a magnet close to the tube while it's energized and we align it so that its field is roughly normal to the surface of the vacuum chamber. If we compute qv x B, we will find that the force due to the magnetic field is perpendicular to the paths of the electrons and to the magnetic field (by the definition of a cross product). In other words, the magnet curves the paths of the electrons and this effect is amplified by the duration that the electrons spend in the field. This effect can easily be observed inside of our cathode ray tube if a magnet is present nearby.
I've included a diagram of the mechanical construction of the accelerator to give a rough idea of how everything will work.