As a lawyer, I would say that an electric stove or microwave wouldn't break this rule since no fire is created. Unless you interpret fire to mean heat of any kind.
My understanding is that when electricity first made its appearance there was a big debate as to whether it could be manipulated on the Sabbath. Orthodox Rabbis decided it could not for spiritual rather than technical reasons.
I'm not so sure this ruling will last for the next 25 years. Some think it is an error.