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I got the above question when I came across the following problem.

A current $I_0$=1.9A flows in a long closed solenoid. The wire it is wound of is in a superconducting state. Find the current flowing in the solenoid when the length of the solenoid is increased by $\eta$=5%.

At first I thought that the total energy stored in the solenoid(i.e.$\frac12Li^2$) does not change.But I soon realized that approach is incorrect as in changing the inductance of the solenoid one has to do work hence the energy of the solenoid changes.
After thinking a lot about I looked at the solution.
In the solution, it says that the flux through the solenoid remains constant. Following is the solution: $$\phi=Li=constant$$
But for a solenoid $L=\frac{\mu_0N^2s}{l}$where N is the number of turns, s is the area of cross-section and $l$ is the length of the solenoid.
Now as we can see $L\propto \frac{1}{l} \implies i\propto l $,
so we get $\frac{I_0}{l}=\frac{I}{(1+\eta)l}$.
Hence final current is $I=I_0(1+\eta)$.
This is where I am stuck because of two question:

  1. On changing the length of the solenoid we are probably increasing the number of turns in the solenoid, which is not considered in the solution.
  2. Even if I agree with above approach to solve the problem, then we are changing the current in the solenoid then how does one knows that the flux ($\phi=Li$) linked with solenoid is not changing when both the current and the inductance of it are changing.
    I would be glad if someone explains it to me.
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1 Answer 1

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A closed superconducting solenoid apparently has the ends connected together so that the current can flow without an external power supply. The only way to increase the length without interrupting the flow would be to stretch the turns further apart. Stretching the solenoid would require work. I think we can assume that work done against magnetic forces would go into the energy of the field. Like you, I can see no basis for assuming that the flux is constant.

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