Since there are a couple of questions here, let’s break it down int two:
**Does it matter how long a computer is powered off for?**
Yes, sort of. Some of the electronics inside the computer can hold energy for a brief period of time. These components aren’t enough to actually power the computer, but they can hold a small amount of voltage for a short period of time (usually a few seconds). In the context of restarting your computer for the purposes of troubleshooting an issue, it’s mostly immaterial. I’ve been doing IT work for literal decades. I can count on one hand the number of times that I had to worry about it at this level. There were a couple of unique cases where I had to unplug the device, press the power button to create a load (this immediately drains any power stored within the circuit), then plug it back in.
If you power a device off, count to 10, then turn it back in, that is generally sufficient except for *extremely* rare circumstances.
**Do electronics need rest?**
Most electronic components have no moving parts. The CPU in your computer, for example, has no moving parts on anything above an atomic scale. That is to say, there are electrons moving around, but that is it. As you use them, they heat up, but as long as this heat remains constant, they don’t experience wear at any rate that is relevant for the typical lifespan of a computer.
This is the story for most of the components in your computer. They get warm during operation, but as long as they stay at a steady-ish temperature, they’ll last a really long time. Turning a computer off allows it to cool down. As the components cool down, they shrink a little bit, then they expand again when they warm up. If the components are heated up and cooled down repeatedly, this expansion/contraction can weaken the soldered connections between components and the board they’re connected to. In this way, a computer that is run constantly actually has a pretty good chance at long life.
There are, however, some components that do have moving parts. Hard disk drives, for example, have spinning metal platters and moving parts inside. The more these run the sooner they wear out. However, the drives have mechanisms to shut the drive down when not in use. Most computers use SSD (solid state drives) these days, so this is less of a concern.
Yes.
If too short (probably a second or few), there may be some issues with power (residual charge in capacitors, etc). Generally not an issue with modern PCs. If too long (a few years), the battery saving BIOS settings would run out (not necessarily an issue), also lubricants in all the fans would dry out causing extra power draw and overheating.
There are other issues that happen over time (i.e. capacitors drying out), but they do not necessarily care whether computer is on or off.
If you leave the computer in a completely depowered state for long enough, the main memory (DRAM) can effectively clear by itself without needing to be done manually by the OS.
RAM does not immediately erase itself completely on loss of power. It does so gradually and chaotically, which leads to possibilities of physical side channel attacks to steal secrets from computers (cold boot attack). This is why it’s suggested to shut down a computer and its power supply for a specified amount of time before powering back in, to really ensure the memory is clear.
Your computer is still able to maintain time as it has an auxillary CMOS battery. This lets the BIOS store settings in its own RAM even when powered off, and for the hardware clock to run.
Last time I had to call to Tech Support, when they wanted me to power my system off, they said Leave it off for 30 seconds before I power it back on again. I believe the thinking being that there are some large capacitors in the power supply which might take 30 seconds to complete discharge and therefore parts of the motherboard could still be on, even though you shut it off.
Yes there is a little bit of a difference just in the fact that it takes a few seconds for some capacitors to fully discharge. More to the point though you may just want to do a restart instead of shutting down and powering back on. They operate a bit differently now and [this article ](https://computer.howstuffworks.com/restarting-shutting-down-computer.htm) can explain it better than I could.
There are already some good answers. I will add another possibility. If the computer is behaving badly because it is overheating, then you will want to turn it off long enough for it to cool off. Also, you should try to relieve the heat stress (clean vents, put it in an area with better circulation, etc.).
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