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You can attach a wire to an object in Earth orbit and use the magnetic field of the Earth to change that object's orbit over time, as well.

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That's an interesting idea, I wonder if you would need a loop or a winding method to get enough counter-emf?

There are at least some proposals that use ionised particles in the earth's upper atmosphere as a conductive elements, meaning you don't need a loop as such- just a wire pointing towards the earth with a big metal ball on the end. Lorentz force means you can accelerate prograde or retrograde.

I find it fascinating to use the natural things you have available as a motive force, and it's really cool to think all you might need is a few bits of conductor and the earth's own relatively static magnetic field to move a satellite around.

I wish humans were less interested in the noisy, explosive things like engines and rockets, and more into subtlety.

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