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how far should these two wires remain from each other as they travel to ground?
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It doesn't matter. You can run them under the same staples if you want to.
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There is a copper wire that runs from the house directly into the ground.
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This is the perfect location. Use a split nut (like the phone installer used) for each new ground wire to be attached to the main ground wire (probably a #4 or #6 copper) unless you can find a split nut that is manufacturer-rated for multiple wires. Simple split nuts should be available in the electrical section of any home center. Here's an example from Home Depot's site: