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Since Mars is only weakly magnetized does Mars not suffer from ground induced currents or are they far worse than on Earth?

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There is no net internal magnetic field to perturb, so how could one induce currents?

The effect arises on Earth because some transient event (e.g., coronal mass ejection or CME) hits Earth's magnetosphere and causes a sudden $\tfrac{dB}{dt}$ which, from Faraday's law, results in induced electric fields. These induced electric fields then drive currents in things like power grids (typically long-distance wires running east-west at high latitudes), oil pipelines, etc.

The only changing magnetic field would be that of the solar wind and the small, localized magnetic anomalies on the surface of Mars. However, these fields are incredibly small so the $\tfrac{dB}{dt}$ would be even smaller.

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