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I have the length of the first coil, the number of turns in both, the current through the first coil and the cross-sectional area of the coil inside. I want to find the induced voltage. I know I should use Faraday's law, but I'm not sure exactly how. I think I've over-thought it and confused myself!

I tried finding the magnetic field of both then summing them together, but this doesn't really help. I've considered flux, but I can't see how I would use this to get an answer. I just don't quite understand how the coils interact with each other in this orientation.

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  • $\begingroup$ Just calculate the B-field generated inside the first, larger coil by a current running through it. You can easily find the formula for that on the internet. Then just use that B-field to calculate the magnetic flux captured per turn of the inner coil. From that you can calculate the voltage induced in the inner coil for a given dI/dt ramp rate of the current through the outer coil. Nothing too difficult. Just need to sit down and work out all the equations and numbers. $\endgroup$
    – user93237
    Commented May 21, 2017 at 21:19

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Here is a simulation done integrating numerically the Biot-Savart law, for a solenoid inscribed into the red rectangle.

            enter image description here

As you can see, the magnetic field inside the solenoid is pretty much uniform.

Thus, once you compute the magnetic field generated by the inner coil, you can easily compute the concatenated flux.

$$\Phi_c=B A$$

where A is the cross section area. Then you just need to apply the Faraday law to get the induced potential.

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