Whichever one of these adapters you end up using:
It is the AxB numbers you need to search for when looking for an extension lead, as they describe the outer and inner diameters of the plug on the end
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You don't need to solder it, though it makes for the neatest job in my opinion. Buying a set of crimp ferrules and the crimp tool will serve the purpose too; they're like very small pipes, you strip and push the wires into the ferrule then crush it gently
Some tips:
When you cut the wire you'll find 2 cores. Cut these cores to different lengths so that whatever you use join the wire (solder, crimps) don't lie side by side. Cut the mating side to the same offsets but the other way round. This makes the end result neater and slimmer and considerably reduces risk of short circuit. Especially for very thin wires I recommend you bend the wire over and wrap the wire core around the outside of the sheath away from the bend, and crimp the whole lot, copper and plastic sheath. This will give support to the joint rather than requiring the copper core to take all the strain of any movement. Do not crush heavily as it's easy to cut the wire inside the ferrule by crushing too much
Fit heat shrink running to the wires before you assemble then move into place and shrink with a hairdryer afterwards
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If you're soldering, same idea about ofsettings the joints. Skin the wire to expose the core and neaten up/twist stands together if they have splayed apart. Using a hot soldering iron, touch it to one wire only and apply a bit of solder to the iron next to the wire so capillary action makes it stick to the wire. Remove it soon as overheating the wire will shrink and damage the sheath
Do the same to the other exposed cores. You now have all wire tips covered in a minute amount of solder but none are joined. Hold one tip so it is adjacent the wire it must join and touch them both with the soldering iron til you see the solder melt and flow, joining the wires, then move the iron away and hold the joint in place till you see a colour change in the solder indicating it has cooled and solidified again.
Doing it this way is a lot easier as you never need more than two hands holding things, which is simpler than trying to get tiny wires to stay adjacent while you solder them together in one hit, straight from being bare copper
If using heat shrink remember to apply unshrunk to the wire first, and move it far enough away from the join that heat conducting down the wire doesn't shrink it. I tend to use heat shrink on each core to be joined and another larger piece over the whole joint