Mars lacks a substantial magnetic field; as a result, the solar wind ablates the Martian atmosphere, making the surface uninhabitable. Therefore, any terraforming attempt will require an artificial Martian magnetic shield. The fundamental challenge of building an artificial magnetosphere is to condense planetary-scale currents and magnetic fields down to the smallest mass possible. Superconducting electromagnets offer a way to do this. However, the underlying physics of superconductors and electromagnets limits this concentration. Based upon these fundamental limitations, we show that the amount of superconducting material is proportional to $B_c^{-2}a^{-3}$, where $B_c$ is the critical magnetic field for the superconductor and $a$ is the loop radius of a solenoid. Since $B_c$ is set by fundamental physics, the only truly adjustable parameter for the design is the loop radius; a larger loop radius minimizes the amount of superconducting material required. This non-intuitive result means that the intuitive strategy of building a compact electromagnet and placing it between Mars and the Sun at the first Lagrange point is unfeasible. Considering reasonable limits on $B_c$, the smallest possible loop radius is $sim$10 km, and the magnetic shield would have a mass of $sim 10^{19}$ g. Most high-temperature superconductors are constructed of rare elements; given solar system abundances, building a superconductor with $sim 10^{19}$ g would require mining a solar system body with several times $10^{25}$ g; this is approximately 10% of Mars. We find that the most feasible design is to encircle Mars with a superconducting wire with a loop radius of $sim$ 3400 km. The resulting wire diameter can be as small as $sim$5 cm. With this design, the magnetic shield would have a mass of $sim 10^{12}$ g and would require mining $sim 10^{18}$ g, or only 0.1% of Olympus Mons.
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