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Default Standard PIR - 12V?

I need to know if the standard PIR you get on a halogen 240V external
light can be put onto a 12V light? I know you can buy the external PIRs
on their own (rated at 1000W for example) but I just want to connect
one side to a 12v battery and the other side to some 12v lighting.

Also, say a lamp is 10W 12V, then this draws 0.833A. If a battery was
3000mAH, would this last for 3.6 hours? Am I correct?

Thanks for any help.

Mat G
Birmingham, UK