Hello, I think the weapon of your choice would be a so called 'micrometer' 
Micrometer - Wikipedia
Plus: If you want to measure in such small dimensions, be aware of the fact that even the warmth of your hand and also the atmosphere around the wire has an impact on the extension of the wire. I'm going with vasquo's oppinion.
//edit: With what I want to say: If you're going to measure a wire by applying voltage, it will expand slightly due to warming up slightly (electrons colliding inside the copper), so I assume you can basically forget that...