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Consider a solenoid having $N$ turns and cylindrical length, a superconducting ring is placed inside having its plane perpendicular to axis of solenoid, if ring (having self-inductance $L$) is very small we can consider $r<<R$ of solenoid and one can assume magnetic field to be going inside the ring to be uniform, but will the inductance of solenoid get changed decrease/increaae by how much. My question is what happens such that inductance change happens how to show it?

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    $\begingroup$ Is this homework? Then please tag it accordingly. $\endgroup$
    – oliver
    Feb 14, 2022 at 15:41
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    $\begingroup$ I mean i just need to understand the details about what happens in the process and how the inductance decrease of increase , calculation i can do myself $\endgroup$
    – Orion_Pax
    Feb 14, 2022 at 16:15

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A superconducting loop as you described it, will deflect all the magnetic flux directed on its $πr^2$ loop area, around the loop towards the inner walls of the larger radius R solenoid. The squeezed flux in a smaller volume inside the solenoid area will therefore increase the strength $B$ of the field around the superconductive loop inside the solenoid around the location where the superconductive loop is.

Through the sc loop area there is no flux going through therefore there is no magnetic field $B$ on the surface area of the sc loop. It will disturb the field in that area and make it no homogeneous. However the flux $Φ$ through the total inner volume of the solenoid will remain unchanged for a given current I of the solenoid for the reasons explained previously.

The inductance of the solenoid is given by:

$$L=\frac{\mu N^{2} A}{\ell}$$

where $l$ the length of the solenoid and $A$ its cross-sectional area and μ the magnetic permeability of the medium inside the solenoid. In your case the sc loop will increase the permeability in that area around the sc loop concentrating the flux towards the inner wall of the solenoid and acting like ferromagnetic core inserted. Thus, increasing locally the inductance of the coil in that location. Effectively, it would be like adding more turns $N$ overall to the solenoid. The excluded area $dA$ from the sc loop is not a problem since no magnetic flux is lost but just diverted around the sc loop $B=\mu n l$.

Expect, with a sc loop inserted at the center of the solenoid to increase the overall inductance $L$ of the solenoid since it acts like a ferromagnetic core or ferrite inserted.

Notes: I assumed that your question assumes that the superconductive loop has already a persistent current $I_{sc}$ flowing inside prior the solenoid is powered up by an external emf source? Therefore the Meissner effect:

MESEFF

If this is not the case and you are referring to first an inactive sc loop material inserted in an inactive solenoid and then powering up the solenoid, then the situation changes and magnetic flux will pass through the sc loop normally and will be trapped there even after the solenoid is powered down.

scltrB

In this second case, concerning the overall inductance of the solenoid for $r<<R$ nothing will really change.

source: http://web.pdx.edu/~pmoeck/lectures/312/supercon.pdf

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  • $\begingroup$ Sir isnt the superconducting property is that it will not allow magnetic field lines to go from the inside of the ring material at r distance ? It will go all other ways just it will not go inside the material , consider a r' radius circle ( r'<r) there js no material there so should nt the field lines pass through it ? $\endgroup$
    – Orion_Pax
    Feb 14, 2022 at 18:58
  • $\begingroup$ Sir there is no mention of such current in problem , also i think just current is there in the solenoid itself . And Sir ring is perpendicular to axis of symmetry $\endgroup$
    – Orion_Pax
    Feb 14, 2022 at 19:07
  • $\begingroup$ @Orion_Pax Answer was updated. In any case a persistent current will be created $I(sc)$ flowing around the sc loop. This is how superconductors work and interact with external magnetic fields. Even by inserting an inactiv sc loop inside an inactive solenoid after the solenoid is powered up the $ΔΦ$ change of magnetic flux created will generate a persistent current in the sc loop. $\endgroup$
    – Markoul11
    Feb 14, 2022 at 19:23
  • $\begingroup$ Sir i got it the thing which i am not getting was inside the ring there is so much space why the field not go there ? i can understand field not go there in case of sphere bc the material is fully blocked with a inside world inside the sphere and one outside but this is not the case with ring it fully open sided isnt Sir? $\endgroup$
    – Orion_Pax
    Feb 14, 2022 at 20:28
  • $\begingroup$ Let me ask as a new question its better $\endgroup$
    – Orion_Pax
    Feb 15, 2022 at 2:20

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