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1

No Earth satellite has the fuel to reach the moon. You would have to go haul it yourself--and you're getting either a broken satellite or one that's out of fuel. You'll do better to launch your own.

5

In the case where the spacecraft can achieve relativistic speeds and no gravity is involved, $\Delta v$ is actually $c$ times the maximal change of rapidity which can be achieved. So the maximal speed the spacecraft can accelerate to, starting from zero, is $c \tanh \frac{\Delta v}{c}$. For example, if the maximal rapidity is $22$, then the maximal speed is \$...

7

@tfb is correct: this is another form of ion propulsion, or generally, electric propulsion. A good general reference for electric propulsion is Fundamentals of Electric Propulsion: Ion and Hall Thrusters from JPL's DESCANSO series. The problem with electric propulsion using ultra-high exhaust velocities is the power required to drive the exhaust beam. If ...

5

That was a good description of a magnetic sail, which is a concept that is out there but hasn't been tested yet. It can change the direction of thrust by changing the orientation of the magnetic field. It has a theoretical advantage over solar sails in that the weight increases with the circumference of the sail rather than with the area, but I guess I ...

3

Let me put this in a different frame. First recall that magnetic fields also store magnetic energy: $$E_{mag} = \int_V H. B\, dV$$ Now, from Lagrangian mechanics, note that: $$L = U-V = E_{mag}+E_{grav}-\frac{mV^2}{2}-\frac{\omega^TI\omega}{2}$$ And let's forget some dissipative effects and the angular terms for now. Recalling Euler-Lagrange's ...

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