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Self-inductance of a circuit is defined as $L=\frac{\Phi}{I}.$ But it is not clear what $\Phi$ (read phi) is. Is it net magnetic flux or flux due to the magnetic field created due to current in the circuit?

Now suppose we have a loop, as shown in the first fig, through which some non-zero current $I$ is passing. A magnetic field will be created and flux through any surface bounded at the circuit will be non-zero, in this case. Hence the inductance of the coil is $L=\frac{\Phi}{I}\not=0.$

enter image description here

Now suppose if a machine creates such a magnetic field so that net magnetic flux (due to the field of the coil and the machine) through the same circuit be zero. Then if $\Phi$ is net magnetic flux the inductance of the coil will be $L=0.$ enter image description here

But it can be found in all books that self-inductance is dependent on geometry. But here the geometry of the circuit and even the current through it is not changed but still, its inductance changed but it shouldn't. This happens because I took $\Phi$ to be net magnetic flux, but if I take it to be flux due to the magnetic field of the circuit only then no problem arises. But I can't be sure of it, since it is not explicitly mentioned anywhere I checked. So am I correct?

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. . . flux through any surface bounded at the circuit will be zero . . . is not true.
If you look at your diagram there is a “flow” (fuux) through the area as shown by the arrows on the magnetic field lines.

The flux $\Phi=\int \vec B \cdot d\vec A$ is thus not zero.

As for geometry do you not think that a loop with one turn will have the different total flux passing though it as a solenoid of the same area but having $100$ turns with the same current passing through it?

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  • $\begingroup$ 1) Sorry, I wanted to write non-zero there. 2) Yes, but I think you misinterpreted my statement "Now suppose if I create such a magnetic field so that flux through the same circuit be zero..." I will edit it for clarity. Sorry for the inconvenience. $\endgroup$
    – Osmium
    Feb 21, 2022 at 14:35
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    $\begingroup$ @Osmium not to be flippant but your question is really along the lines of what if some gravitational masses did not always attract but sometimes repelled. Magnetic fields are generated by currents, the field exists whether you measure it with an inductive loop or not. If there is no flux in your loop then there is no induction but the inductance of the loop is a geometric quantity telling you how would the flux induce a current if there was one. It is like volume of a jar, even if you do not fill it with water it still has a volume, a geometric quantity. $\endgroup$
    – hyportnex
    Feb 21, 2022 at 15:03

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