Force Between Two Concentric Solenoids

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The discussion centers on calculating the magnetic force between two concentric solenoids, where the inner solenoid is influenced by the time-varying current in the outer fixed solenoid. The inner solenoid, which can slide in and out, experiences induced current due to the magnetic field generated by the outer solenoid. Participants explore whether this magnetic force can be calculated analytically or if it requires numerical methods for a solution. The problem can be simplified by considering both solenoids as thin shells, allowing for a clearer understanding of their interaction. Ultimately, the feasibility of calculating the magnetic force between the two solenoids is the primary focus of the discussion.
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Imagine a finite length solenoid with outer radius R1 and inner radius R2. This solenoid has a time-varying current going though it. This solenoid is also fixed so that it cannot move. Now imagine another solenoid, coaxial with the first, with its outer radius equal to R2 such that it can slide in and out of the first solenoid. The ends of this solenoid are connected together and no current is being externally applied to it. The only current going through it is the induced current from the first coil. My question is, would it be possible to calculate the magnetic force on this solenoid? Or is this a problem that would need to be solved numerically?
 
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My question can be simplified so that both solenoids are a thin shell instead of having a finite thickness. I'm just wondering how one solenoid within another reacts.
 

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