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Why does he need to wear a Faraday cage to crawl along the lines, though? Surely if he's not touching the ground, no harm can come to him. The current isn't going to go through him when it could go through the metal wire instead, in the same way that birds sit on the line without any current passing through them. I assume the thing he does of touching a metal rod from the line to the helicopter removes the potential difference between them. Maybe the Faraday cage is just in case something goes wrong at this stage. Obviously I didn't pay enough attention in high-school physics.
Why the faraday cage suit? Because when the voltages are high enough, you get corona effects in a field around the main conductor. The concern isn't so much the voltage gradient to ground, but simply the gradient within the corona itself. What you don't want to have happen is to have a voltage potential between your two hands, because your heart is in between, and a few miliamps of current are enough to mess up the heartbeat pattern. The faraday suit ensures that any voltage differences around the body are carried in it's very-conductive wire mesh, and not in the moderately-conductive bloodstream of the wearer. Remember, "It's the volts that jolts, but it's the mils that kill".