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Does anyone know why it is that a stronger external magnetic field will result in a greater force on a current-carrying conductor exposed to this field? I'm struggling to find an answer to this that actually explains the physics behind it, rather than just relating it back to the formula F=BIlsintheta.

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In a sense, this cannot be explained. It is just an observation of how nature works. We can describe it, but not explain it.

One can come upon this phenomenon as a consequence of other principles, for example, the Biot-Savart law, or from Maxwell's equations. But none of these are explanations. You can swim upstream looking for precursor principles to the phenomenon, but you will swim for a long time and never find a starting point. At some point you stop swimming and decide that you have reached a principle that you will simply have to accept without explanation.

Maybe someday we will know why things are the way they are. But I doubt it.

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