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In Sadiku, he used the formula $dλ=dΨ *\frac{\text { Ienclosed}}{I}$ to determine the total flux linkage for coaxial cable for $ρ<a$ and for $a<ρ<b$, but I applied this formula for the solenoid and it didn't work, the way that works for the solenoid is by using $λ=N*Ψ$.

So why we multiply by $\frac{\text { Ienclosed }}{I}$in the coaxial cable?

Here is the exercice for reference

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  • $\begingroup$ Which book is this from? It looks like the often shown but misleading derivation of $L′_{in}$, self-inductance of a straight wire per unit length that is due to field inside the wire only. This is a dubious concept, because the actual self-inductance per unit length, taking into account all magnetic flux, comes out infinite (due to too much magnetic flux in infinity). Real wires have finite self-inductance per unit length due to fact they have finite dimensions. $\endgroup$
    – Ján Lalinský
    Commented May 27, 2023 at 17:39
  • $\begingroup$ It's from Sadiku Elements of Electromagnetics. I'm just confused as when to multiply by Ienc/I or just N*I $\endgroup$
    – Elie Makdissi
    Commented May 27, 2023 at 21:52
  • $\begingroup$ It's based on the bizarre idea that in calculations of self-inductance, magnetic flux for any given loop in the wire cross-section has to be weighed by the amount of current going through that loop. The first occurence of this I know is in paper by Rosa g3ynh.info/zdocs/refs/NBS/Rosa1908.pdf . This weighing method covers only part of self-inductance, due to flux inside the wire, and gives contribution to self-inductance consistent with magnetic energy stored in the wire $\int_{wire} \frac{1}{2\mu_0}B^2 dV$. $\endgroup$
    – Ján Lalinský
    Commented May 27, 2023 at 22:53
  • $\begingroup$ The concept is flawed because only total self-inductance $L$ or self-inductance per unit length $dL/d\ell$ can be measured. This is affected by magnetic flux outside the wire as well. Thus contribution due to flux in the wire is only intermediate calculation result, not the real self-inductance. For infinite wire, $dL/d\ell$ comes out infinite. Finite result can be obtained if the wire is finite - Rosa does this calculation and gets self-inductance that is superlinear in $\ell$, thus self-inductance per unit length is not $\frac{\mu_0}{8\pi}$, but depends on $\ell$. $\endgroup$
    – Ján Lalinský
    Commented May 27, 2023 at 22:56
  • $\begingroup$ Thus the sentence by Sadiku "eqs. (8.11.1) and (8. 11.2) are also applicable to finding the inductance of any infinitely long straight conductor of finite radius." is incorrect. $\endgroup$
    – Ján Lalinský
    Commented May 27, 2023 at 23:08

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