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I am trying to understand flux linkage but so far none of the explanations have made sense to me. Most all material on flux linkage is discussed in terms of a coil but here I am thinking of a thick conducting wire and I don’t see how it generalizes.

Suppose we take a small section of the conductor illustration

Then the flux through this small piece is

$$ d\phi_x = \frac{\mu_0 I x}{2\pi r^2} dx $$

Now most resources I have looked at say that the entire cross sectional area does not enclose this flux, thus we have to multiply it my the fraction $x^2/r^2$. That part makes no sense to me since it seems like the cross-sectional area does include that flux.

I have also read some explanations which say that we are only taking into account a portion $\pi x^2/\pi r^2$ of the total current which is why that is the flux linkage factor. This doesn’t make sense to me since it seems we have already taken that into account In calculating the magnetic field and we obtain the total current through integration.

Could someone explain to me how these two points of view on the flux linkage are correct? I know there are some similar questions about flux linkage but they did not sufficiently address these issues to me.

Thank you!

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    $\begingroup$ I didn't understand this either. No explanations on the internet. Perhaps the curriculum is false? They probably have been copy-pasting this since decades $\endgroup$ Commented Jun 11, 2023 at 15:32

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Maybe the expression written is for the flux density from the center of the circle. Therefore, for that particular tubular section, the fractional value is multiplied.

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  • $\begingroup$ This does not provide an answer to the question. Once you have sufficient reputation you will be able to comment on any post; instead, provide answers that don't require clarification from the asker. - From Review $\endgroup$
    – Miyase
    Commented May 11 at 18:26
  • $\begingroup$ Yeah, and there was I thinking, people with knowledge have an open mind. Thank you $\endgroup$ Commented May 12 at 6:20
  • $\begingroup$ It isn't a matter of having an open mind, it's simply the rules of this site: when you're posting in the answer section, you're expected to provide a full-fledge answer. $\endgroup$
    – Miyase
    Commented May 12 at 7:27

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