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Yes, but this condition is such an extreme one that:

  • you must include the internal resistance of the battery when calculating the current
  • you must include the resistance of all the wires when calculating the current
  • it is unlikely you can find a 1V battery with high enough AHr rating to do this for more than a tiny moment of time before it dies
  • a resistor having a value of 0.001 ohms is usually just called a wire
  • 1000 amperes of current, if you do get it, will melt all standard gauges of wire (even 0000 the largest, which is almost 1/2 an inch in diameter); to get a "wire" that can safely conduct this much current it will need to be several inches in diameter!!!

The only way you could do it exactly as you describe in your question is with perfect ideal components, which just do not exist in the real world.

But if redesigned to allow for the nonzero resistances of real components, using a low internal resistance voltage source (but probably not a battery), using large diameter copper bars (not wires) that can safely carry 1000 amperes, and getting permission from your power company to connect directly into a substation for power (not on a standard home or industrial distribution line); you can do something similar.

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Q: Is it possible to generate current of 1000 AMPS using a 1V Battery and resistor of value 0.001ohms?
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