??? talking about “stable” arc in air, "0.125 to 48 in, the arc voltages averaged 34 V/in " (0.32 to 122 cm, 13.4 V/cm)
That’s 3 volts at a 10th of an inch. (And you can weld with a 12V solar panel). I think there is a lower limit, a width below which you can’t maintain an arc in air, because of the ionization behaviour of air, but it’s at less than 3V / eighth of an inch.
You may be correct. But the equations suggest a lower limit of about 20 V with a 1 mm width.
I have always wondered about the definitive answer to this question.
So, back to the OP. I’ve noticed that if you buy brand new GFCI receptacles, the receptacles have a far more solid feel than the $0.60 home depot special receptacles. It takes a lot more insertion force, and once you plug the device in, there’s a lot more force needed to pull the plug back out. The plug won’t rattle as much, either.
So as far as I can tell, there’s only 2 reasons the OP might have this “half plugged in” problem. Either it’s worn out receptacles (I replaced a half dozen in my house this year when I went around doing the GFCI upgrade), or something has too short of a cord.
Either needs replacing. Worn out receptacles don’t make just “half contact”. Many times they are in such poor condition there’s 1/10 the contact area or it’s just hanging on by an electrical arc. I found old receptacles where if you plug something in, it’ll just hang loose and you have to jiggle the plug to even get power to the device at all. Terrible, and I bet there’s a series arc in there sometimes when this happens.
Just buy the slightly higher grade this time. Get the $3-$5 commercial grade receptaclesor at least the “Preferred” residential grade that cost about $1.50. The commercial grade ones are a whole different quality tier, they are far heavier and you can just tell they mean serious business.
Oh, one note : if you do decide to replace some faulty receptacles, you should replace with tamper resistant receptacles. Those can save small children from shock when they play with paperclips (I did this and while I didn’t die, I could have) and you should also replace some of them with GFCI if the place doesn’t have it. The GFCI ones need to be upstream and they can protect everything downstream if wired in correctly. (the instruction packet that comes with each receptacle basically tells you how to do it)
One final note : if you go around replacing receptacles, make sure the power is off! Plug a test device into the receptacle (a lamp or radio) and make sure it comes on. (so you know the test device isn’t faulty). Turn off/on breakers until the device turns off. (a radio helps if you don’t have an assistant). Leave that breaker off!
For extra paranoia, check with a voltage tester or I put a screwdriver blade between the hot wire and ground. Better to get an arc flash than to get shocked.
I think you will find that the voltage drops in the region 1mm - 4mm. (Or rises in the region 4mm-1mm…)
Also, it depends on the gas (air), humidity, confinement, surface material.
And, regarding the OP, arcing across a surface (tracking) is different again.
Hell no, electrical arcs are the number one source of car and truck fires.
Excellent find, thank you! So I take back what I said. Well, sort of… I am still pretty sure you can’t get a sustained arc at 12 V to 14 V. But you can certainly get an intermittent/sputtering arc at those voltage levels which will produce damage.
(As a side note, virtually all my experience in arc-related failures is for systems operating at 115 V / 400 Hz or 270 VDC. The latter in particular can produce some very impressive arcs. And sustainable ones at that. I don’t have any experience w/ automotive arcs.)
So we do agree that the OP needs to replace his or her receptacles. And stop using the broken ones until replacement, right? If the plug won’t stay in, it’s hanging loose. The blade is not necessarily halfway in contact with the receptacle metal, the very tip could be in contact.
Since it’s loose in the receptacle, it’s possible for it to just barely make contact well enough to get current moving, and then once the flow is established, if the plug moves just enough, an electrical arc will form. Probably series since probably just 1 prong will be loose enough. And then there’s heating and maybe a fire.
On a day to day basis, you won’t see that. It’s a statistical phenomenon. Out of every bad receptacle with a loose plug in the USA, only a small percent start fires at any moment. Still not a safe situation.
120 VAC receptacles are very cheap and very easy to replace. If you suspect there is any problem with a receptacle, you should simply replace it ASAP.
At lower voltages you require a higher current to sustain an arc in air. When you are doing arc welding, if your electrode gets too close to the material, your arc will go out when your power supply doesn’t supply the increased current required… This will indead give you a sputtering arc: the power in the arc is the arc voltage times the arc current.
If you have enough current, it is indeed possible to weld at 6V. Aother problem then is, if the arc is short, it doesn’t spread very much, and you weld a narrow hot spot, which is not what you want.
Well it helps when you know and have worked with Peter on numerous occasions. 
Interesting to read about the causes of fires in US Trucks. When I was involved in British haulage, I remember seeing a similar report about European trucks. In our case, electrical failures were at the bottom of the list; the most common cause being overheated tyres. This was most commonly due to underinflation or binding brakes. The second most likely cause was a mechanical failure in the engine; usually related to turbochargers.
Electrical fires, where they did occur were frequently due to the drivers overloading the in-cab circuits with their own equipment. It seems that otherwise sensible people will, when faced with a fuse that keeps blowing, put a higher rated one in, or even use aluminium foil.