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Suppose we have a rectangular current-carrying loop with current $i$, then we know that the magnetic force on each side can be found as:

$$F=iL\times B$$

Where $L$ is the vector in the direction of the current with the length of the corresponding side. On the other hand we know that if there are $N$ turns instead of just one this will affect the torque, since it'll also affect the magnetic dipole moment. However what about force? Do this also affect the force?

My intuition tells that it'll affect the force and that it'll become $F=N(iL\times B)$, however I'm unsure if I'm correct and I also don't know how to prove it. Is this result correct? How do we prove it if it is?

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Yes, what you wrote is right. Each loop feels force from the magnetic field and the loop cannot exert net force on itself, so your expression is the only option.

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