To my knowledge there are no perfect constant current sources. It will take on whatever voltage is necessary to do so including both positive and negative voltages. I'll give you a few examples to consider:ฤก) The current source is a mathematical construct that maintains a constant current. The current source is a different animal. With more current and a fixed voltage there will be more power. As you stated as the resistance is decreased more current flows. Examples include a battery or wall outlet. In my opinion the voltage source is relatively straightforward. CAUTION - this is an answer from an electrical engineer.
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