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My understanding is that if you have a current, I then 2I will have an average electron drift velocity that is twice that of I. Now, to produce 2I from I with uniform resistance requires a voltage that is also twice the original, V and 2V respectively.

Kinetic Energy is 1/2mv^2. An two-fold increase in speed will produce a four-fold increase in energy. How is it then that double the potential energy will produce four-times the kinetic energy?