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If an electroscope is charged negatively the electrons on the leaves will repell each other and stand apart. It is clear than there is a steady force between the leaves that counters gravity. How is this force explained in terms of string theory or quantum field theory and invoking the concept of virtual particle exhange between electrons lodged on the leaves?

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you don't need quantum theory to understand that, just classical EM theory as explain in any Jackson volume is more than enough. All those quantum theories are just used to provide corrections which are entirely unobservable in such apparatus as you propose

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