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    $\begingroup$ This is not an answer to your question, but you know that Ohm's law it's not a fundamental law. While this law describes many (or most) cases, some materials/systems may not follow this relation $\endgroup$ Commented Dec 2, 2016 at 18:40
  • $\begingroup$ @cinico i know non ohmic resistors do not follow this relation but hows this related to the question. $\endgroup$ Commented Dec 2, 2016 at 18:44
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    $\begingroup$ As I said, it's not the answer to your question but a side note that I thought it would be interesting to add because there are materials/systems with a "negative resistance". To be more specific, this doesn't mean that you will have a negative electrical current, but only that the current can decrease with increasing voltage. $\endgroup$ Commented Dec 2, 2016 at 19:33
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    $\begingroup$ So what you've discovered might be described as "current needs to go inversely with the voltage if you want to keep the same power output" as well as "current needs to go proportional to voltage if you want to keep the same effective resistance." Therefore what you have discovered is that there is no way to simultaneously modify both current and voltage in a way that keeps both the same: that is what the contradiction tells you. So if you raise the voltage and keep the resistance the same, you're going to be forced to increase both current and power dissipated. $\endgroup$ Commented Dec 2, 2016 at 20:53
  • $\begingroup$ I don't see any contradiction in what you describe. You changed the variables between the two equations hence anything can happen since you have some "free variable" that can make things turn out how you want. In this case: when you increase voltage you can just increase the power and tada, the current can increase too! $\endgroup$ Commented Dec 3, 2016 at 11:03