So far so good. I began to do the rest of the math the hard way, then I found this:
http://www.mpoweruk.com/solar_power.htmwhich gives me a shortcut. The world's energy needs might be met, with ideal efficiency and ideal placement (not including significant inherent losses in transport and storage) with a total of 118,000 km sq. area of solar panel. The US has 1/17 of the world's population, but has 5 times the world average energy expenditures per person. So:
118,000 sq. km x 5/17 = 34,706 sq km of solar panels to live at our current affluence level. Remember line losses and storage are very, very significant too. So you can double or triple that area. We'll double it. About 70,000 sq km will be needed, minimally. Note:
that's the size of South Carolina. Meanwhile, the materials needed, the area of land covered, and the fact that you're extracting heat from the atmosphere will have a significant environmental impact. Not to mention the fact that you're crowding out plants from the best sunlight on the planet, competing with them in their niche. Whenever humankind has competed head-to-head, it has always won. So you could expect more extinctions. That's going to have a negative environmental effect, too. And remember, the population will grow, as it always has when we've had a plentiful source of energy.
Though as I think about it, extracting heat from the atmosphere right now would not at all be bad. But it's generally a trade-off just the way any other energy source is. This is not going to "free" mankind. Two things will free mankind: either reduce the population or solve the problem of interstellar space travel.
I hope you see why I'm skeptical.
PS added with edit: you apparently didn't know what I meant by "efficiency" if you began to tell me about differential sunlight. I meant how much of the sunlight shining on a panel can be converted into electricity. The best ones we have are 8% efficient. Second Law of Thermodynamics says you're not going to get near 100% of it. Seventy percent is cited as the theoretical limit .
As for NASA beaming us electricity from space, first, what would be the environmental impact of it? Beaming it in the microwave spectrum is not going to be environmentally friendly. Second, that might end up being less, not more efficient than gathering it on earth, simply because you gather it once with 8% efficiency beam it down, and then have to gather it again with 8% efficiency, and probably you have loses storing it twice, instead of once. (I know the numbers might not be that low, but since you're multiplying them, that doesn't matter that much.
Just because somebody has an exciting idea doesn't mean it'll work.