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I am reading Ampere's original work. Therein he describes an experiment called "equilibrium of opposite current experiment".

In this experiment, two immobile straight wires with current flowing in opposite directions are placed near each other. Then a mobile astatic coil is placed near them. It experiences zero torque.

Next he describes an experiment called "equilibrium of bent wire experiment".

In this experiment, an immobile bent wire and straight wire with current flowing in opposite directions are placed near each other. Then a mobile astatic coil is placed near them. It experiences zero torque.

Then he concludes that magnetic effects of bent wire in "equilibrium of opposite current experiment" is equivalent to magnetic effects of straight wire in "equilibrium of bent wire experiment". This is fine.

But he also concludes (maybe without proof) that magnetic effects on bent wire in "equilibrium of opposite current experiment" is equivalent to magnetic effects on straight wire in "equilibrium of bent wire experiment". Why is this so?

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[The electro]magnetic effects on bent wire in "equilibrium of opposite current experiment" is equivalent to [electro]magnetic effects on straight wire in "equilibrium of bent wire experiment". Why is this so?

Because Ampère was opposed to any violations of Newton's Third Law.

See §20.3 (pp. 261-262) of:

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