I am thinking about mounting a 1000W 120V wall heater on an exterior wall, in my bathroom. Assuming I have room to do so, would it be possible to put rigid insulation behind the heater's mounting box? Could this be a fire hazard?
Yes, it will be a fire hazard. The smoke when it burns is toxic, the reason fire-fighters first started using oxygen tanks. IRC requires it be covered for fire safety in an attic or crawlspace under most conditions....
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17,000 dryer fires a year, when did you last clean the inside of the dryer near motor or the exhaust ducting?