Yes, I believe they will.
The metal body is essentially a faraday cage. The electronics are shielded to prevent interference with the radio. Of course everything is dependent on how close. Full on EMP 5 miles away - likely nothing does.
The reason the grid will not is all the power lines are antennas.
Having worked just a little bit with mu metal shielding to re-direct RF, one thing I learned is the effectiveness of protection varies widely. Optimally, you require metal to be a complete enclosure with no air gaps at all. But that alone usually will not provide enough shield in many situations. Multi-layers of complete shielding like a Russian Nesting Doll with each layer completely air tight sealed, and non-magnetic non-conducting spacing between each layer will work well in most applications. Think attenuation, then you want minimum coupling of what gets through, to be hitting the next layer, and so on, and so on. The first metal cage in some ways has a weakness in that it does create a less energetic coupling of a time delayed ( phase shifted ) magnetic pulse to it's next layer. So spacing helps to minimize that transfer of energy.
When I explained this to someone once, they said, " Then I should store my generator inside a small metal trash can, and put that inside a larger metal trash can."
I said, yes, but you should maintain good spacing of all areas between those cans. Like two-by-four's under the smaler can so it does not directly touch the outer can.
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Btw, one X-Military officer once told me if the missles are ever inbond, he's going outside to sit and watch, because if it hitts, he would want to be gone in an instant.
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After the US set off the largest ever blast, after one second the fireball was 5 miles wide, after one minute the fireball was 60 miles wide.
These levels of energy are beyond the understanding of most. The lucky would be dead beford they knew anything was happening.
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