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This web article says:

Astatic needles are basically two needles of approximately the same magnetic strength, mounted on top of one another. They are parallel, but their magnetic poles are on opposite sides, as shown in the diagram. This way, when inside (any) external magnetic field, the torque exerted on one needle is equal and opposite to the torque exerted on the other. Since they are connected, the system overall remains unaffected by the field.

Thus it seems to me that astatic needles should also be immune to other magnetic fields. Why is it not so?

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The astatic needles should be immune to other uniform magnetic fields. The web article you refer to describes the use of a pair of these needles with a galvanometer but one of the needles is inside the coil and one is outside the coil (as shown in the initial photographs). The direction of the field inside and outside the coil is opposite. Because the polarities of the two needles are opposite, putting them in fields in the opposite direction will result in two torques in the same direction.

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