Your understanding of old tube-type televisions is correct! The setup is indeed that you have a cathode, which is an electron gun, which fires electrons at the front of the TV. At the front, you have a series of materials which will glow in red, green, or blue when they are hit by the electrons, and by firing electrons in the right places, you get a full color image.
One of the problems with using this method to provide wireless power is that cathodes are kind of huge. You must remember how bulky and unwieldy an old cathode ray tube monitor or TV was. Compare a CRT TV with a new LCD display – the LCD wins both in terms of image quality and in the size of the TV. LCD displays are almost hilariously thin in comparison to the old style TVs.
If you just want to power up your phone again, setting up a rather delicate glass vacuum chamber that can’t get much smaller than it is, and aiming it at your phone is less than ideal. So, we’ve developed other methods of transferring power wirelessly, but not using cathodes. If you have an electric toothbrush, the charger for it uses one of these alternate methods, and Ikea is starting to use the same system built into some of their end tables, so you can charge your phone overnight by just setting it in the right place – no wires required.
This method uses the current from the wall (i.e., moving electrons) to induce a magnetic field by running the current through a coil. A coil within the object to be charged can convert the magnetic field of the charger back into a current, thereby charging the device/phone/toothbrush. The limitation here is we have designed these objects to create very small magnetic fields, so that in order to use them you need to be really close to them. This is fine for most purposes, since it mostly requires us to put our devices in a specific location, and it’s still more convenient than physically plugging it into the wall.
Of course, the other issue with firing electrons across the room to charge something, is that firing charged particles across a physical space where people are frequently walking is a super bad idea.
There are two other contexts in which you might encounter high speed electrons, and they’re in radioactive decay and in medical procedures such as radiation therapy. These electrons are typically 10 to 100 times more energetic than the electrons fired out of a cathode, but the danger to a human’s health is a combination of energy and the amount of time you spend being exposed to it. Radiation therapy is highly calibrated to the part of the body they’re aiming at, and the length of time you spend under it, and can be a very effective treatment for treating certain cancers. Radioactive decay is a little less controlled, and while the human body can tolerate high energy doses for a little while, in general you want to minimize the amount of time you spend hanging around nuclear waste.
We’ve sort of internalized this dosage calculation already – if you go outside, spending 10 minutes in direct sunlight is probably not enough to burn you (unless you are very pale), but if you spend an hour out there, your chance of sunburn has increased dramatically. The UVA & UVB light we get from the sun has an energy which is about 5,000 times less than the electrons fired from a cathode, so we can guess that our cathode ray beam is a bit more dangerous than sitting outside, but significantly less dangerous than sitting next to some unstable uranium. (If you are now worried about sitting in front of an old TV – don’t be; these numbers are for the electrons produced by the cathode. In the TV, all of them are absorbed to produce the image, and so the TV screen itself produces very little radiation.) The real issue is that if you put an electron gun in homes, the length of time you spend being exposed to that beam of electrons is really, really hard to control. Say you put it next to your desk and want to charge your stuff while you’re working. How long do you spend in front of your desk? Or do you charge it while you sleep? Do you keep everything in a Faraday cage far from your bed, or do you normally keep your phone on an end table next to you?
It may be technically possible, but for any kind of wireless power, we’re going to have to make sure that the methods can safely integrate into our homes, without adding radiation worries to our daily lives. This is why the magnetic loop method is so appealing – it doesn’t require any kind of charged particle to be beamed through open space, and without the receiving wire loop on the device you’ll want to charge, there’s no transfer of electricity, so you’ll never shock or burn yourself on it.
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