I recently encountered a question which went like this,

A current-carrying loop is placed in a plane perpendicular to a long current carrying wire at a distance from it. What is the force acting on the loop?

The answer was given to be zero for any shape of the loop.

I couldn't find a convincing reason for this and it left me puzzling.

Is the answer right? If yes, how and if not, why?

Integrating all over the loop is fine for a regular shape but the question is about any shape of the loop.

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    $\begingroup$ Draw a diagram... $\endgroup$ – user207455 Jul 6 at 16:52
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    $\begingroup$ The force will be proportional to $\oint{d\vec{s}\times\vec{B}}$. See math.stackexchange.com/questions/2654912/… for how to relate this to the divergence of $\vec{B}$, which is zero. $\endgroup$ – G. Smith Jul 6 at 17:27
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    $\begingroup$ Proofs for arbitrarily shaped loops always rely on Stokes’ Theorem. The trick is realizing that when you write the cross product using $\epsilon_{ijk}$, you can interpret the integral as a line integral of the scalar product of $d\vec{s}$ with something, and then apply Stokes! $\endgroup$ – G. Smith Jul 6 at 17:52

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