The key here is that magnetic fields will bend the path of a moving charged particle, and we can make use of this effect to control a beam. Crucially for the Accelerate! recipe, you need a larger magnetic field to bend a faster-moving particle.
In the cathode ray tube, electrons are ejected from the cathode and accelerated through a voltage, gaining some 600 km/s for every volt they are accelerated through. Some of these fast-moving electrons crash into the gas inside the tube, causing it to glow, which allows us to see the path of the beam. Helmholtz coils can then be used to apply a quantifiable magnetic field by passing a known current through them.
A magnetic field will cause a force to act on the electrons which is perpendicular to both their direction of travel and the magnetic field. This causes a charged particle in a magnetic field to follow a circular path. The faster the motion of the particle, the larger the circle traced out for a given field or, conversely, the larger the field needed for a given radius of curvature of the beam. Making this quantitative point is impossible without control over both particle energy and magnetic field, so this will need to be stated if your demo doesn’t have both of these.
In the case of the CRT TV, the paths of the electrons are distorted by the magnet being brought near the screen. The picture on the screen is dependent on the electrons precisely hitting phosphors on the back of the screen, which emit different colours of light when impacted. The electrons are thus forced to land in the wrong place, causing the distortion of the image and the psychedelic colours.
Adapted from: Oxford Dept of Physics
A description of the CRT: National Electronics 2BP1
Using the relationship of Voltage and Work per charge, we can determine the velocity of the electrons as they hit the screen.
By varying the electric or magnetic fields of the 'control' plates, images are able to be drawn on a phosphorescent screen.
A CRT uses an accelerating potential of 5200. V. What velocity do the electrons have when they pass through the anode? 4.3x10^7 m/s